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Fuzzy regression approach to modelling transfer moulding for microchip encapsulation
Journal of Materials Processing Technology 140 (2003) 147–151
Fuzzy regression approach to modelling transfer moulding for microchip encapsulation K.W. Ip, C.K. Kwong∗ , Y.W. Wong Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China
Abstract Transfer moulding is one of the popular processes to perform microchip encapsulation for electronic packages. Existing analytical models, such as generalised Hele–Shaw model, seem to be inadequate to model the process accurately in real world environment due to the complex inter-relationships among the encapsulant properties, process conditions, mould design parameters and overall moulding performance and the inherent fuzziness of the moulding systems. It is quite often that the observed values from the transfer moulding for microchip encapsulation may not be regular. Although statistical regression method could be used to perform the modelling, high degree of fuzziness inherent in transfer moulding systems for microchip encapsulation makes the obtained models having wide possibility range. In this paper, the fuzzy regression concept and its application in modelling transfer moulding for microchip encapsulation are described. Fuzzy regression is a well-known method to deal with the problems with a high degree of fuzziness. Thirty-two experiments were firstly 8−2 experimental plan in this study that involved eight process parameters and three quality characteristics. The conducted based on an 2iv experimental settings and results of the 30 experiments were then used to develop three fuzzy linear regression models, which relate various process parameters and the three quality characteristics, respectively. With the use of these models, proper process conditions and prediction range of individual quality measures can be obtained. Two validation tests were carried out to evaluate the developed models. Results of the tests show that the actual values of all the quality measures were found within the corresponding prediction ranges. The calculated prediction errors for the three output measures were all less than 5%. © 2003 Elsevier B.V. All rights reserved. Keywords: Process modelling; Fuzzy regression; Transfer moulding; Microchip encapsulation
1. Introduction The package of microchip devices is the housing that protects the chip from mechanical stress, environmental damage and electrostatic discharge during further handling. Among the existing encapsulation methods such as transfer moulding, dispensing and potting, transfer moulding commands the largest share of production volume that accounts for more than 90% of market share because of its high performance/cost ratio [1]. Although transfer moulding is a commonly used method for microchip encapsulation, determination of process parameters setting for satisfactory quality parts is difficult and is usually based on the experience and intuition of engineers. The process is more difficult than injection moulding because of the complex behaviour of encapsulant during moulding, the presence of microchips and wires in the mould cavities and the inherent ∗ Corresponding author. Present address: Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China. Tel.: +852-27666610; fax: +852-23625267. E-mail address: [email protected] (C.K. Kwong).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00702-7
fuzziness of transfer moulding systems. A proper process model of transfer moulding, therefore, is critical to provide a fundamental understanding of the relationships between the various input and output parameters and, in turn, is helpful in determining a proper setting of process parameters. Analytical modelling method is one of the common ways to model many manufacturing processes. Process parameters can be determined from analytical models in very short time. However, due to the complexity of the mould filling process and the difficulty in obtaining an accurate rheological description of the moulding compound being processed, a comprehensive analytical model describing the actual filling process of transfer moulding is not yet available. In fact, microchip encapsulation process is a theoretically three-dimensional, transient, reactive problem with moving resin front. To simplify the analytical model of this complicated problem, many studies considered that the flow of moulding compound is governed by the generalised Hele–Shaw model. The model assumes an incompressible viscous polymeric resin fills the mould cavity under the non-isothermal conditions and symmetric thermal boundary conditions. It neglects the velocity component
148
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
in the thickness direction of the mould cavity. This velocity component is important for analysing the defects, such as paddle shift and leadframe deformation, as well as for predicting the degree of cure of the thermosetting compound. Numerous research works have been conducted on numerical simulation to perform the flow modelling of microchip encapsulation and predict the mould defects, such as voids, flash, incomplete fill and warpage etc. [2–4]. These works aimed at providing engineers information to identify potential moulding problems and evaluated alternative designs without physically building moulds and going through moulding trials, which are costly and time-consuming. Nguyen [2] demonstrated the benefits of reactive flow simulation in transfer moulding of electronic packages from which mould layouts and process conditions could be properly determined. Useful information of layout designs, leadframe configurations and material mouldability characteristics could also be obtained. Apart from the analytical and numerical modelling methods, statistical regression and artificial neural networks (ANNs) have also been used in modelling various manufacturing processes such as injection moulding [5] and grinding [6]. Artificial neural networks do not require a priori assumption of the functional form of the model and derive their knowledge of the process from examining sets of input data and their corresponding outputs, which makes it an effective method for modelling complex non-linear processes. However, as the knowledge is represented as numeric weights in ANNs, the rules and reasoning process are not readily explainable. ANNs model the process in a ‘black box’ operation in which the relationships among various parameters and the process behaviour cannot be known. Statistical regression method is the most common modelling method used in the manufacturing industry. The regression models are found to be accurate in the range that the models are developed. In other words, statistical regression models can be applied only if the given data are distributed statistically. The regression model is constructed using a probability model and the relationship between independent and dependent variables is crisp. It makes rigid assumptions about the statistical properties of the model. These assumptions, as well as the aptness of the linear regression model, are difficult to justify unless a sufficiently large data set is available. The violation of such basic assumptions could adversely affect the validity and performance of statistical regression. In contrast, fuzzy regression is possibilistic in nature and fuzzy functions are used to represent the relationship between dependent and independent variables. One advantage of using fuzzy regression analysis is that it can process the fuzzy sample data such as (xi , Yi ), where Yi is a fuzzy number and xi is the vector of the explained variables, in a way that is closer to the reality [7]. In this paper, fuzzy regression approach to modelling transfer moulding for microchip encapsulation is described. This research could be treated as the first
work to investigate fuzzy regression approach to process modelling.
2. Fuzzy linear regression Fuzzy linear regression was first introduced by Tanaka et al. [8], which aims to model vague process using fuzzy functions. These functions were defined by Zadeh’s extension principle that provides a general method for extending non-fuzzy mathematical concepts to deal with fuzzy quantities [9]. Unlike statistical regression modelling that is based on probability theory, fuzzy regression is based on possibility theory and fuzzy set theory. Deviations between observed values and estimated values are assumed to be due to system fuzziness. It has been found that fuzzy regression could be more effective than statistical regression when the degree of system fuzziness is high. Tanaka defined a fuzzy linear regression model as shown below: ˜ 1 x1 + A ˜ 2 x2 + · · · + A ˜ j xj + · · · + A ˜ N xN = A ˜x ˜0 +A y˜ = A (1) where y˜ is the fuzzy output, x = [x1 , x2 , . . . , xN ]T is the ˜ = real-valued input vector of independent variables and A ˜ 0, A ˜ 1, A ˜ 2, · · · , A ˜ N ] is a vector of the model’s fuzzy pa[A ˜ j are represented in the rameters. The fuzzy parameters A form of symmetric triangular fuzzy numbers denoted by ˜ j = (αj , cj ), j = 0, 1, . . . , N with its membership funcA tion as shown below: |α − aj | 1− j , αj − cj ≤ aj ≤ αj + cj , cj µA˜ j (aj ) = 0, otherwise where aj is the centre value of the fuzzy number and cj the spread. Hence, the fuzzy linear regression model can be rewritten as follows: y˜ = (α0 , c0 ) + (α1 , c1 )x1 + (α2 , c2 )x2 + · · · + (αN , cN )xN The estimated output y˜ can be obtained by using the extension principle [10]. The derived membership function of the fuzzy number y˜ is |y − αT x| 1− , x = 0, cT |x| µy˜ (y) = 1, x = 0, y = 0, 0, otherwise where |x| = (|x1 |, |x2 |, . . . , |xN |)T , the central value of y˜ is αT x, and the spread (range) of y˜ is cT |x|. ˜ j = (αj cj ), the folTo determine the fuzzy coefficients A lowing linear programming (LP) problem is formulated:
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
Fig. 1. Fuzzy regression model.
Minimize
J=
N j=0
subject to
N
cj
M
|xij |
i=1
αj xij + (1 − h)
j=0 N
N
cj |xij | ≥ yi ,
j=0
αj xij − (1 − h)
j=0
N
cj |xij | ≤ yi
j=0
cj ≥ 0, αj ∈ R, j = 0, 1, 2, . . . , N, xi0 = 1, i = 1, 2, . . . , M, 0 ≤ h ≤ 1
(2)
where J is the total fuzziness of the fuzzy regression model. The h value, which is between 0 and 1, is a threshold level to be chosen by the decision maker. This term is referred to as a degree of fitness of the fuzzy linear model to its data. Constraints in (2) mean that each observation yi has at least h degree of belonging to y˜ as µy˜ i (yi ) ≥ h (i = 1, 2, . . . , M). Therefore, the objective of solving the LP problem is to de˜ j such that the total vaguetermine the fuzzy parameters A ness J is minimised subject to µy˜ i (yi ) ≥ h (i = 1, 2, . . . , M). The fuzzy regression by solving the above LP problem is illustrated in Fig. 1. The relationship between the model and samples is shown in the figure. It is noted that the fuzzy regression contains all samples within its range. This indicates that it expresses all possibilities, which the samples embody and exist for the system under consideration. 3. Development of fuzzy regression models for transfer moulding of electronic packages Transfer moulding is one of the popular processes to perform microchip encapsulation. In this process, the thermoset moulding compound (typically a solid epoxy reform) is preheated and then placed into the pot of the moulding tool. A transfer cylinder, or plunger, is used to inject the moulding compound into the runner system and gates of the mould. The moulding compound then flows over the chip, wire-bonds and leadframes, encapsulating the microelectronic device. In this study, a comparatively new method
149
for microelectronic encapsulation, liquid epoxy moulding (LEM), is investigated. LEM is classified as a type of transfer moulding. The moulding process is very similar with that of conventional transfer moulding, but the epoxy moulding compound used is liquid in nature. The relatively low viscosity of liquid epoxy moulding compound allows low operating pressures and minimises problems associated with wire sweep. The in-mould cure time for the liquid epoxy resin is no longer than that for current transfer moulding compounds. LEM is a highly complex moulding process, which involves more than 10 influential material properties, mould design parameters and process parameters in process design. Although modelling the LEM process is critical to understand the process behaviour and optimise the process, the process is much difficult to characterise due to the complex behaviour of epoxy encapsulant and the inherent fuzziness of the moulding system such as inconsistent properties of epoxy moulding compound and environmental effect on moulding temperature. In current practice, a time-consuming and expensive trial-and-error approach is adopted to qualify a mould design or to optimise the process conditions for a given mould. In the following, development of a process model of LEM process for microchip encapsulation using fuzzy regression is described. To develop a fuzzy regression model for relating process parameters and quality characteristics of LEM, significant process parameters and quality characteristics have to be identified first. Eight significant process parameters, which are named as independent variables were identified and their operating values are shown below: • • • • • • • •
Top mould temperature (150, 180 ◦ C), x1 . Bottom mould temperature (150, 180 ◦ C), x2 . Filling time (30, 60 s), x3 . Transfer force (90, 100 kgf), x4 . Curing time (30, 50 s), x5 . 1st injection profile (−1 mm/2 s, −1 mm/4 s), x6 . 2nd injection profile (−7 mm/2 s, −7 mm/4 s), x7 . 3rd injection profile (−8 mm/2 s, −8 mm/4 s), x8 .
Three quality characteristics (named as dependent variables) were also identified as shown below: • Wire sweep (%), y. • Void (mm2 ), z. • Flash (mm2 ), w. Thirty-two experiments were carried out in which 30 sets of the data and results were used to develop the fuzzy regression models of LEM and two sets of them were used to validate the developed models. The fuzzy linear regression model for dependent variable y (wire sweep) can be represented as follows: y = A0 + A1 x1 + A2 x2 + A3 x3 + A4 x4 + A5 x5 + A6 x6 + A7 x7 + A8 x8 where Ai (i = 0–8) are fuzzy parameters
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K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
Table 1 The centre and width values of fuzzy parameters for wire sweep Fuzzy parameters
Centre
A0 A1 A2 A3 A4 A5 A6 A7 A8
9.8907 −0.0267 0.0094 0.0623 −0.1481 0.1447 −0.8301 −0.1084 1.3133
Table 2 Experimental settings for the validation tests Width 0.0000 0.0218 0.0134 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
The LP model can be formulated as follows: 30 8 cj |xij | Minimize J = j=0
subject to
8
i=1
αj xij + (1 − h)
8
j=0
j=0
8
8
αj xij − (1 − h)
j=0
cj |xij | ≥ yi , cj |xij | ≤ yi
j=0
cj ≥ 0, αj ∈ R, j = 0, 1, 2, . . . , 8, xi0 = 1,
Setting (◦ C)
Top mould temperature, x1 Bottom mould temperature, x2 (◦ C) Filling time, x3 (s) Transfer force, x4 (kgf) Curing time, x5 (s) 1st injection profile, x6 (mm/s) 2nd injection profile, x7 (mm/s) 3rd injection profile, x8 (mm/s)
y = (9.8907, 0.0000) + (−0.0267, 0.0218)x1 + (0.0094, 0.0134)x2 + (0.0623, 0.0000)x3 + (−0.1481, 0.0000)x4 + (0.1447, 0.0000)x5
Test
1 2
Centre
Range
5.6787 5.6787
2.8523 1.5884
−2.8264 to 8.5310 −4.0903 to 7.2671
+ (0.0048, 0.0000)x6 + (0.0053, 0.0000)x7 + (−0.0062, 0.0000)x8
Error (%)
2.7000 1.6000
5.6400 0.7250 3.1825
Table 4 Results of validation tests for void Test
1 2
Fuzzy regression Spread
Centre
Range
0.0464 0.0557
0.0063 0.0099
0.0000–0.0527 0.0000–0.0656
Actual value (mm2 )
Error (%)
0.0065 0.0103
3.0800 0.8800
Average error
3.4800
Table 5 Results of validation tests for flash Test
Average error
+ (−0.0011, 0.0000)x4 + (0.0005, 0.0000)x5
Actual value (%)
Average error
+ (1.3133, 0.0000)x8
+ (−0.0003, 0.0000)x2 + (0.0003, 0.0000)x3
150 180 60 100 30 1/2.00 7/2.00 8/2.00
Spread
1 2
z = (0.0392, 0.0000) + (0.0005, 0.0003)x1
150 180 60 90 30 1/2.00 7/2.00 8/2.00
Fuzzy regression
+ (−0.8301, 0.0000)x6 + (−0.1084, 0.0000)x7 With the same calculation, the fuzzy linear regression models for dependent variables z (void) and w (flash) can be obtained, respectively:
Test 2
Table 3 Results of validation tests for wire sweep
i = 1, 2, . . . , 30, 0 ≤ h ≤ 1 With the threshold level h = 0.1, the above LP model was solving using MATLAB and the central value and width of each fuzzy parameter were obtained as shown in Table 1. Thus, the fuzzy linear regression model for the dependent variable y (wire sweep) is shown below:
Test 1
Fuzzy regression Spread
Centre
Range
0.0689 0.0689
0.0312 0.0423
0.0000–0.1001 0.0000–0.1112
Actual value (mm2 )
Error (%)
0.0305 0.0454
2.2951 6.8282 4.5617
4. Model validation In order to evaluate the developed models, two validation tests were carried out, where the experimental setting is shown in Table 2. As can be seen from the results (Tables 3–5), all actual values of measurements are found to fall within their corresponding prediction ranges. The average prediction error of flash is found to be the greatest, whereas the prediction error of void is the lowest.
w = (0.0221, 0.0000) + (0.0004, 0.0003)x1 + (0.0003, 0.0002)x2 + (0.0000, 0.0000)x3 + (−0.0005, 0.0000)x4 + (−0.0013, 0.0000)x5 + (0.0004, 0.0000)x6 + (−0.0080, 0.0000)x7 + (0.0053, 0.0000)x8
5. Discussion and conclusion This work modelled the LEM process with fuzzy regression analysis. From the developed regression models, it can be found that among the eight process parameters, the top
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
mould temperature is the most significant factor to induce the fuzziness of the moulding system. It is also found that the prediction error of flash is the highest among the three quality measures, which may be due to the fact that it is difficult to obtain the measurement values of flash accurately. Therefore, the measurement error of flash would be larger. The ratio of outliners of flash becomes higher than that of wire sweep and void, respectively. With the use of the 30 sets of experimental data and results, fuzzy linear regression approach was introduced to develop the process models for LEM in which the independent variables of mould temperature, filling time, transfer force, curing time and the injection profile are investigated. Two validation tests have been performed to evaluate the effectiveness of the process models. The predicted values of the quality measures have the average deviation less than 5%. Fuzzy linear regression could be applied to process modelling by fuzzifying the statistical linear regression model with a fuzzy linear function and fuzzy parameters. Currently, the system used only triangular membership functions for the fuzzy parameters. Further work would investigate the use of non-triangular non-symmetric membership functions in fuzzy parameter setting and an optimisation technique for obtaining the global optimal quality characteristics. Furthermore, more systematic methods could be explored in identifying the outliners and thus diminishing their influence to the process model.
Acknowledgements The work described in this paper was supported substantially by a grant from the Hong Kong Polytechnic University
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(project number: H-ZH35). The authors would like to thank Kras corporation for their kind assistance in conducting the experiments and the validation tests of this research.
References [1] L.S. Turng, Computer-aided engineering (CAE) for the microelectronic packaging process advances in computer-aided engineering (CAE) of polymer engineering, ASME MD 49 (1994) 191– 207. [2] L.T. Nguyen, Reactive flow simulation in transfer molding of IC packages, in: IEEE Proceedings of the 43rd Electronic Components and Technology Conference, 1993, pp. 375–390. [3] L.S. Turng, V.W. Wang, On the simulation of microelectronic encapsulation with epoxy molding compound, J. Reinf. Plast. Comp. 12 (1993) 506–519 [4] F.J. Lim, L.T. Nguyen, W.V.D. Bogert, Simulation of the transfer molding process for encapsulation of integrated circuits, in: Proceedings of the 62nd Annual Meeting of Society of Rheology, 1990. [5] G.H. Choi, K.D. Lee, N. Chang, Optimization of process parameters of injection molding with neural network application in a process simulation environment, CIRP Ann. 43 (1) (1994) 449– 452. [6] E. Brinksmeier, H.K. Tonshoff, C. Czenkush, C. Heinzel, Modeling and optimization of grinding processes, J. Intell. Manuf. 9 (1997) 303–314. [7] B. Wu, N.F. Tseng, A new approach to fuzzy regression models with application to business cycle analysis, Fuzzy Sets Syst. 130 (2002) 33–42. [8] H. Tanaka, S. Uejima, K. Asai, Fuzzy linear regression model, in: Proceedings of the International Congregation on Applied Systems and Cybernetics, 1980, pp. 2933–2938. [9] K.J. Kim, H. Moskowitz, M. Koksalan, Fuzzy versus statistical linear regression, Eur. J. Oper. Res. 92 (1996) 417–434. [10] H.J. Zimmermann, A. Jones, A. Kaufmann, Fuzzy Sets Theory and its Applications, Reidel, Boston, 1986.
Fuzzy regression approach to modelling transfer moulding for microchip encapsulation K.W. Ip, C.K. Kwong∗ , Y.W. Wong Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China
Abstract Transfer moulding is one of the popular processes to perform microchip encapsulation for electronic packages. Existing analytical models, such as generalised Hele–Shaw model, seem to be inadequate to model the process accurately in real world environment due to the complex inter-relationships among the encapsulant properties, process conditions, mould design parameters and overall moulding performance and the inherent fuzziness of the moulding systems. It is quite often that the observed values from the transfer moulding for microchip encapsulation may not be regular. Although statistical regression method could be used to perform the modelling, high degree of fuzziness inherent in transfer moulding systems for microchip encapsulation makes the obtained models having wide possibility range. In this paper, the fuzzy regression concept and its application in modelling transfer moulding for microchip encapsulation are described. Fuzzy regression is a well-known method to deal with the problems with a high degree of fuzziness. Thirty-two experiments were firstly 8−2 experimental plan in this study that involved eight process parameters and three quality characteristics. The conducted based on an 2iv experimental settings and results of the 30 experiments were then used to develop three fuzzy linear regression models, which relate various process parameters and the three quality characteristics, respectively. With the use of these models, proper process conditions and prediction range of individual quality measures can be obtained. Two validation tests were carried out to evaluate the developed models. Results of the tests show that the actual values of all the quality measures were found within the corresponding prediction ranges. The calculated prediction errors for the three output measures were all less than 5%. © 2003 Elsevier B.V. All rights reserved. Keywords: Process modelling; Fuzzy regression; Transfer moulding; Microchip encapsulation
1. Introduction The package of microchip devices is the housing that protects the chip from mechanical stress, environmental damage and electrostatic discharge during further handling. Among the existing encapsulation methods such as transfer moulding, dispensing and potting, transfer moulding commands the largest share of production volume that accounts for more than 90% of market share because of its high performance/cost ratio [1]. Although transfer moulding is a commonly used method for microchip encapsulation, determination of process parameters setting for satisfactory quality parts is difficult and is usually based on the experience and intuition of engineers. The process is more difficult than injection moulding because of the complex behaviour of encapsulant during moulding, the presence of microchips and wires in the mould cavities and the inherent ∗ Corresponding author. Present address: Department of Applied Physics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, PR China. Tel.: +852-27666610; fax: +852-23625267. E-mail address: [email protected] (C.K. Kwong).
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00702-7
fuzziness of transfer moulding systems. A proper process model of transfer moulding, therefore, is critical to provide a fundamental understanding of the relationships between the various input and output parameters and, in turn, is helpful in determining a proper setting of process parameters. Analytical modelling method is one of the common ways to model many manufacturing processes. Process parameters can be determined from analytical models in very short time. However, due to the complexity of the mould filling process and the difficulty in obtaining an accurate rheological description of the moulding compound being processed, a comprehensive analytical model describing the actual filling process of transfer moulding is not yet available. In fact, microchip encapsulation process is a theoretically three-dimensional, transient, reactive problem with moving resin front. To simplify the analytical model of this complicated problem, many studies considered that the flow of moulding compound is governed by the generalised Hele–Shaw model. The model assumes an incompressible viscous polymeric resin fills the mould cavity under the non-isothermal conditions and symmetric thermal boundary conditions. It neglects the velocity component
148
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
in the thickness direction of the mould cavity. This velocity component is important for analysing the defects, such as paddle shift and leadframe deformation, as well as for predicting the degree of cure of the thermosetting compound. Numerous research works have been conducted on numerical simulation to perform the flow modelling of microchip encapsulation and predict the mould defects, such as voids, flash, incomplete fill and warpage etc. [2–4]. These works aimed at providing engineers information to identify potential moulding problems and evaluated alternative designs without physically building moulds and going through moulding trials, which are costly and time-consuming. Nguyen [2] demonstrated the benefits of reactive flow simulation in transfer moulding of electronic packages from which mould layouts and process conditions could be properly determined. Useful information of layout designs, leadframe configurations and material mouldability characteristics could also be obtained. Apart from the analytical and numerical modelling methods, statistical regression and artificial neural networks (ANNs) have also been used in modelling various manufacturing processes such as injection moulding [5] and grinding [6]. Artificial neural networks do not require a priori assumption of the functional form of the model and derive their knowledge of the process from examining sets of input data and their corresponding outputs, which makes it an effective method for modelling complex non-linear processes. However, as the knowledge is represented as numeric weights in ANNs, the rules and reasoning process are not readily explainable. ANNs model the process in a ‘black box’ operation in which the relationships among various parameters and the process behaviour cannot be known. Statistical regression method is the most common modelling method used in the manufacturing industry. The regression models are found to be accurate in the range that the models are developed. In other words, statistical regression models can be applied only if the given data are distributed statistically. The regression model is constructed using a probability model and the relationship between independent and dependent variables is crisp. It makes rigid assumptions about the statistical properties of the model. These assumptions, as well as the aptness of the linear regression model, are difficult to justify unless a sufficiently large data set is available. The violation of such basic assumptions could adversely affect the validity and performance of statistical regression. In contrast, fuzzy regression is possibilistic in nature and fuzzy functions are used to represent the relationship between dependent and independent variables. One advantage of using fuzzy regression analysis is that it can process the fuzzy sample data such as (xi , Yi ), where Yi is a fuzzy number and xi is the vector of the explained variables, in a way that is closer to the reality [7]. In this paper, fuzzy regression approach to modelling transfer moulding for microchip encapsulation is described. This research could be treated as the first
work to investigate fuzzy regression approach to process modelling.
2. Fuzzy linear regression Fuzzy linear regression was first introduced by Tanaka et al. [8], which aims to model vague process using fuzzy functions. These functions were defined by Zadeh’s extension principle that provides a general method for extending non-fuzzy mathematical concepts to deal with fuzzy quantities [9]. Unlike statistical regression modelling that is based on probability theory, fuzzy regression is based on possibility theory and fuzzy set theory. Deviations between observed values and estimated values are assumed to be due to system fuzziness. It has been found that fuzzy regression could be more effective than statistical regression when the degree of system fuzziness is high. Tanaka defined a fuzzy linear regression model as shown below: ˜ 1 x1 + A ˜ 2 x2 + · · · + A ˜ j xj + · · · + A ˜ N xN = A ˜x ˜0 +A y˜ = A (1) where y˜ is the fuzzy output, x = [x1 , x2 , . . . , xN ]T is the ˜ = real-valued input vector of independent variables and A ˜ 0, A ˜ 1, A ˜ 2, · · · , A ˜ N ] is a vector of the model’s fuzzy pa[A ˜ j are represented in the rameters. The fuzzy parameters A form of symmetric triangular fuzzy numbers denoted by ˜ j = (αj , cj ), j = 0, 1, . . . , N with its membership funcA tion as shown below: |α − aj | 1− j , αj − cj ≤ aj ≤ αj + cj , cj µA˜ j (aj ) = 0, otherwise where aj is the centre value of the fuzzy number and cj the spread. Hence, the fuzzy linear regression model can be rewritten as follows: y˜ = (α0 , c0 ) + (α1 , c1 )x1 + (α2 , c2 )x2 + · · · + (αN , cN )xN The estimated output y˜ can be obtained by using the extension principle [10]. The derived membership function of the fuzzy number y˜ is |y − αT x| 1− , x = 0, cT |x| µy˜ (y) = 1, x = 0, y = 0, 0, otherwise where |x| = (|x1 |, |x2 |, . . . , |xN |)T , the central value of y˜ is αT x, and the spread (range) of y˜ is cT |x|. ˜ j = (αj cj ), the folTo determine the fuzzy coefficients A lowing linear programming (LP) problem is formulated:
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
Fig. 1. Fuzzy regression model.
Minimize
J=
N j=0
subject to
N
cj
M
|xij |
i=1
αj xij + (1 − h)
j=0 N
N
cj |xij | ≥ yi ,
j=0
αj xij − (1 − h)
j=0
N
cj |xij | ≤ yi
j=0
cj ≥ 0, αj ∈ R, j = 0, 1, 2, . . . , N, xi0 = 1, i = 1, 2, . . . , M, 0 ≤ h ≤ 1
(2)
where J is the total fuzziness of the fuzzy regression model. The h value, which is between 0 and 1, is a threshold level to be chosen by the decision maker. This term is referred to as a degree of fitness of the fuzzy linear model to its data. Constraints in (2) mean that each observation yi has at least h degree of belonging to y˜ as µy˜ i (yi ) ≥ h (i = 1, 2, . . . , M). Therefore, the objective of solving the LP problem is to de˜ j such that the total vaguetermine the fuzzy parameters A ness J is minimised subject to µy˜ i (yi ) ≥ h (i = 1, 2, . . . , M). The fuzzy regression by solving the above LP problem is illustrated in Fig. 1. The relationship between the model and samples is shown in the figure. It is noted that the fuzzy regression contains all samples within its range. This indicates that it expresses all possibilities, which the samples embody and exist for the system under consideration. 3. Development of fuzzy regression models for transfer moulding of electronic packages Transfer moulding is one of the popular processes to perform microchip encapsulation. In this process, the thermoset moulding compound (typically a solid epoxy reform) is preheated and then placed into the pot of the moulding tool. A transfer cylinder, or plunger, is used to inject the moulding compound into the runner system and gates of the mould. The moulding compound then flows over the chip, wire-bonds and leadframes, encapsulating the microelectronic device. In this study, a comparatively new method
149
for microelectronic encapsulation, liquid epoxy moulding (LEM), is investigated. LEM is classified as a type of transfer moulding. The moulding process is very similar with that of conventional transfer moulding, but the epoxy moulding compound used is liquid in nature. The relatively low viscosity of liquid epoxy moulding compound allows low operating pressures and minimises problems associated with wire sweep. The in-mould cure time for the liquid epoxy resin is no longer than that for current transfer moulding compounds. LEM is a highly complex moulding process, which involves more than 10 influential material properties, mould design parameters and process parameters in process design. Although modelling the LEM process is critical to understand the process behaviour and optimise the process, the process is much difficult to characterise due to the complex behaviour of epoxy encapsulant and the inherent fuzziness of the moulding system such as inconsistent properties of epoxy moulding compound and environmental effect on moulding temperature. In current practice, a time-consuming and expensive trial-and-error approach is adopted to qualify a mould design or to optimise the process conditions for a given mould. In the following, development of a process model of LEM process for microchip encapsulation using fuzzy regression is described. To develop a fuzzy regression model for relating process parameters and quality characteristics of LEM, significant process parameters and quality characteristics have to be identified first. Eight significant process parameters, which are named as independent variables were identified and their operating values are shown below: • • • • • • • •
Top mould temperature (150, 180 ◦ C), x1 . Bottom mould temperature (150, 180 ◦ C), x2 . Filling time (30, 60 s), x3 . Transfer force (90, 100 kgf), x4 . Curing time (30, 50 s), x5 . 1st injection profile (−1 mm/2 s, −1 mm/4 s), x6 . 2nd injection profile (−7 mm/2 s, −7 mm/4 s), x7 . 3rd injection profile (−8 mm/2 s, −8 mm/4 s), x8 .
Three quality characteristics (named as dependent variables) were also identified as shown below: • Wire sweep (%), y. • Void (mm2 ), z. • Flash (mm2 ), w. Thirty-two experiments were carried out in which 30 sets of the data and results were used to develop the fuzzy regression models of LEM and two sets of them were used to validate the developed models. The fuzzy linear regression model for dependent variable y (wire sweep) can be represented as follows: y = A0 + A1 x1 + A2 x2 + A3 x3 + A4 x4 + A5 x5 + A6 x6 + A7 x7 + A8 x8 where Ai (i = 0–8) are fuzzy parameters
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Table 1 The centre and width values of fuzzy parameters for wire sweep Fuzzy parameters
Centre
A0 A1 A2 A3 A4 A5 A6 A7 A8
9.8907 −0.0267 0.0094 0.0623 −0.1481 0.1447 −0.8301 −0.1084 1.3133
Table 2 Experimental settings for the validation tests Width 0.0000 0.0218 0.0134 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
The LP model can be formulated as follows: 30 8 cj |xij | Minimize J = j=0
subject to
8
i=1
αj xij + (1 − h)
8
j=0
j=0
8
8
αj xij − (1 − h)
j=0
cj |xij | ≥ yi , cj |xij | ≤ yi
j=0
cj ≥ 0, αj ∈ R, j = 0, 1, 2, . . . , 8, xi0 = 1,
Setting (◦ C)
Top mould temperature, x1 Bottom mould temperature, x2 (◦ C) Filling time, x3 (s) Transfer force, x4 (kgf) Curing time, x5 (s) 1st injection profile, x6 (mm/s) 2nd injection profile, x7 (mm/s) 3rd injection profile, x8 (mm/s)
y = (9.8907, 0.0000) + (−0.0267, 0.0218)x1 + (0.0094, 0.0134)x2 + (0.0623, 0.0000)x3 + (−0.1481, 0.0000)x4 + (0.1447, 0.0000)x5
Test
1 2
Centre
Range
5.6787 5.6787
2.8523 1.5884
−2.8264 to 8.5310 −4.0903 to 7.2671
+ (0.0048, 0.0000)x6 + (0.0053, 0.0000)x7 + (−0.0062, 0.0000)x8
Error (%)
2.7000 1.6000
5.6400 0.7250 3.1825
Table 4 Results of validation tests for void Test
1 2
Fuzzy regression Spread
Centre
Range
0.0464 0.0557
0.0063 0.0099
0.0000–0.0527 0.0000–0.0656
Actual value (mm2 )
Error (%)
0.0065 0.0103
3.0800 0.8800
Average error
3.4800
Table 5 Results of validation tests for flash Test
Average error
+ (−0.0011, 0.0000)x4 + (0.0005, 0.0000)x5
Actual value (%)
Average error
+ (1.3133, 0.0000)x8
+ (−0.0003, 0.0000)x2 + (0.0003, 0.0000)x3
150 180 60 100 30 1/2.00 7/2.00 8/2.00
Spread
1 2
z = (0.0392, 0.0000) + (0.0005, 0.0003)x1
150 180 60 90 30 1/2.00 7/2.00 8/2.00
Fuzzy regression
+ (−0.8301, 0.0000)x6 + (−0.1084, 0.0000)x7 With the same calculation, the fuzzy linear regression models for dependent variables z (void) and w (flash) can be obtained, respectively:
Test 2
Table 3 Results of validation tests for wire sweep
i = 1, 2, . . . , 30, 0 ≤ h ≤ 1 With the threshold level h = 0.1, the above LP model was solving using MATLAB and the central value and width of each fuzzy parameter were obtained as shown in Table 1. Thus, the fuzzy linear regression model for the dependent variable y (wire sweep) is shown below:
Test 1
Fuzzy regression Spread
Centre
Range
0.0689 0.0689
0.0312 0.0423
0.0000–0.1001 0.0000–0.1112
Actual value (mm2 )
Error (%)
0.0305 0.0454
2.2951 6.8282 4.5617
4. Model validation In order to evaluate the developed models, two validation tests were carried out, where the experimental setting is shown in Table 2. As can be seen from the results (Tables 3–5), all actual values of measurements are found to fall within their corresponding prediction ranges. The average prediction error of flash is found to be the greatest, whereas the prediction error of void is the lowest.
w = (0.0221, 0.0000) + (0.0004, 0.0003)x1 + (0.0003, 0.0002)x2 + (0.0000, 0.0000)x3 + (−0.0005, 0.0000)x4 + (−0.0013, 0.0000)x5 + (0.0004, 0.0000)x6 + (−0.0080, 0.0000)x7 + (0.0053, 0.0000)x8
5. Discussion and conclusion This work modelled the LEM process with fuzzy regression analysis. From the developed regression models, it can be found that among the eight process parameters, the top
K.W. Ip et al. / Journal of Materials Processing Technology 140 (2003) 147–151
mould temperature is the most significant factor to induce the fuzziness of the moulding system. It is also found that the prediction error of flash is the highest among the three quality measures, which may be due to the fact that it is difficult to obtain the measurement values of flash accurately. Therefore, the measurement error of flash would be larger. The ratio of outliners of flash becomes higher than that of wire sweep and void, respectively. With the use of the 30 sets of experimental data and results, fuzzy linear regression approach was introduced to develop the process models for LEM in which the independent variables of mould temperature, filling time, transfer force, curing time and the injection profile are investigated. Two validation tests have been performed to evaluate the effectiveness of the process models. The predicted values of the quality measures have the average deviation less than 5%. Fuzzy linear regression could be applied to process modelling by fuzzifying the statistical linear regression model with a fuzzy linear function and fuzzy parameters. Currently, the system used only triangular membership functions for the fuzzy parameters. Further work would investigate the use of non-triangular non-symmetric membership functions in fuzzy parameter setting and an optimisation technique for obtaining the global optimal quality characteristics. Furthermore, more systematic methods could be explored in identifying the outliners and thus diminishing their influence to the process model.
Acknowledgements The work described in this paper was supported substantially by a grant from the Hong Kong Polytechnic University
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(project number: H-ZH35). The authors would like to thank Kras corporation for their kind assistance in conducting the experiments and the validation tests of this research.
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