Modeling of epoxy dispensing process using a hybrid fuzzy regression approach

Abstract

In the semiconductor manufacturing industry, epoxy dispensing is a popular process commonly used in die-bonding as well as in microchip encapsulation for electronic packaging. Modeling the epoxy dispensing process is important because it enables us to understand the process behavior, as well as determine the optimum operating conditions of the process for a high yield, low cost, and robust operation. Previous studies of epoxy dispensing have mainly focused on the development of analytical models. However, an analytical model for epoxy dispensing is difficult to develop because of its complex behavior and high degree of uncertainty associated with the process in a real-world environment. Previous studies of modeling the epoxy dispensing process have not addressed the development of explicit models involving high-order and interaction terms, as well as fuzziness between process parameters. In this paper, a hybrid fuzzy regression (HFR) method integrating fuzzy regression with genetic programming is proposed to make up the deficiency. Two process models are generated for the two quality characteristics of the process, encapsulation weight and encapsulation thickness based on the HFR, respectively. Validation tests are performed. The performance of the models developed based on the HFR outperforms the performance of those based on statistical regression and fuzzy regression.

Appendix

Tanaka et al. [24] formulated the fuzzy regression problem as the following linear programming problem:

$$ \matrix{ {\text{Minimize}} &{J = \sum\limits_{{j = 1}}^{{{N_{\text{NR}}}}} {\left( {c_{{_0}}^{\prime } + c_j^{\prime }\sum\limits_{{i = 1}}^M {\left| {x_{{_j}}^{\prime }(i)} \right|} } \right)} } \\ }<!end array> $$

(18)

where M is the number of data sets used for training the epoxy dispensing model, and $x_{{_j}}^{\prime }(i)$ is the i-th data set with respect to the j-th transformed variable of the epoxy dispensing model, subject to:

$$ \left( {\alpha_0^{\prime } + \sum\limits_{{j = 1}}^{{{N_{\text{NR}}}}} {\alpha_j^{\prime }\,x_j^{\prime }(i)} } \right) + \left( {1 - h} \right)\left( {c_{{_0}}^{\prime } + \sum\limits_{{j = 1}}^{{{N_{\text{NR}}}}} {c_j^{\prime }\left| {x_{{_j}}^{\prime }(i)} \right|} } \right) \geqslant y(i) $$

(19)

$$ \left( {\alpha_0^{\prime } + \sum\limits_{{j = 1}}^{{{N_{\text{NR}}}}} {\alpha_j^{\prime }\;x_j^{\prime }(i)} } \right) + \left( {1 - h} \right)\left( {c_0^{\prime } + \sum\limits_{{j = 1}}^{{{N_{\text{NR}}}}} {c_j^{\prime }\left| {x_j^{\prime }(i)} \right|} } \right) \leqslant y(i) $$

(20)

$$ {c_j} \geqslant 0,\;\;\;{\alpha_j} \in R,\;{\text{for all }}i, $$

(21)

$$ 0 \leqslant h \leqslant 1,i = 1,2, \cdots M,j = 1,2, \cdots {N_{\text{NR}}}. $$

(22)

J in Eq. (18) is the total fuzziness of the epoxy dispensing model. The value of h in Eqs. (19) and (20) is between 0 and 1. It is referred to as a fitting degree of the epoxy dispensing model to the given data sets and is subjectively chosen by decision makers. Constraints (19) and (20) restrict that the observation of the i-th data set y(i) has at least h degree of belonging to $\widetilde{y}(i)$ as ${\mu_{{\widetilde{y}(i)}}}\left( {y(i)} \right) \geqslant h\left( {i = 1,2, \cdots, M} \right)$. Therefore, the objective of solving the linear programming problem [Eqs. (18), (19), (20), (21), and (22)] was to determine the fuzzy nonlinear parameters $\widetilde{A}_j^{\prime } = \left( {c_j^{\prime },\alpha_j^{\prime }} \right)$ such that the total vagueness J is minimized subject to ${\mu_{{\widetilde{y}(i)}}}\left( {y(i)} \right) \geqslant h\;\left( {i = 1,\;\;2, \cdots, M} \right)$. Therefore, the intervals of the epoxy dispensing process derived from the approach of Tanaka et al. [24] were determined by all the collected data sets and the value h. With the advice from the company supporting this research [18], h = 0.1 was used.