Ful-Chiang Wu

Optimization of robust design for multiple quality characteristics

The Taguchi method has recently been widely applied to variability reduction for increased quality and lower cost in many different industries. The traditional Taguchi method was focused on optimizing a single quality characteristic. A real problem in a product or process possesses multiple quality characteristics. The optimization methods of multiple quality characteristics design have thus become crucial issues for industry. Several articles have presented approaches to optimizing the parameter design with multiple quality characteristics. Few have focused primarily on optimizing the correlated multiple quality characteristics problem. This research presents an approach to optimizing the correlated multiple quality characteristics with asymmetric loss function by a mathematical programming model. The goal is minimizing the total average quality loss for experiments. This proposed procedure is illustrated with data from nine previously published articles. A numerical analysis of the model is provided and the results are compared with those of prior approaches.

Optimization of correlated multiple quality characteristics robust design using principal component analysis

The use of the Taguchi method for improving the design and quality of products and processes has become widespread among different industries. The traditional Taguchi method focused on one characteristic to optimize a combination of parameter conditions. In practice, most products have more than one quality characteristic. The methods of multiple quality characteristics design have become very important for industries. Several studies have presented approaches addressing multiple quality characteristics. Few published articles have focused primarily on optimizing correlated multiple quality characteristics. This research presents an approach to optimizing correlated multiple quality characteristics by using proportion of quality loss reduction and principal component analysis. The results reveal the advantages of this approach in that the optimal parameter design using proportion of quality loss reduction is the same as that using the Taguchi traditional method for one quality characteristic; the chosen optimal design is robust for optimizing correlated multiple quality characteristics.

Robust design of nonlinear multiple dynamic quality characteristics

Robust design is an important method to reduce variation and to improve the performance of products or processes at low cost including static and dynamic quality characteristics. The desirability function approach is an attractive method in industry for the optimization of static multiple quality characteristics. Most of the robust design research in the literature focused on problems with static and linear dynamic quality characteristics. In this paper, we present a method to optimize the nonlinear multiple dynamic quality characteristics based on the double-exponential desirability function. The example of a temperature control circuit design is provided to demonstrate the implementation and usefulness of the proposed method.

A Wavelet-Based Approach in Detecting Visual Defects on Semiconductor Wafer Dies

The objective of this paper is to implement a two-dimensional wavelet transform (2-D WT) approach for detecting visual defects such as particles, contamination, and scratches on semiconductor wafer dies. The gray image of 1/20 of a wafer die is initially processed by smooth and high-pass filters. Then, it is decomposed directly by 2-D WT at multiple scales and different wavelet bases. The interscale ratio from the wavelet transform modulus sum (WTMS) across adjacent decomposition levels (scales) for suspicious pixels on a wafer die is calculated. Since irregular edges in a small domain preserve much more wavelet energy, an edge pixel potentially belongs to a visual defect if its interscale ratio is less than a predefined threshold. The proposed approach is template-free and is easy to implement, so it is suitable for more product varieties and small-batch production. Real wafer dies with synthetic defects are used as testing samples to evaluate the performance of proposed approach. Experimental results from a small amount of testing samples show that the proposed method is able to identify particle, contamination, and scratch defects without missed detection and false alarm by appropriate choice of wavelet bases, scale, and image resolution. The proposed inspection approach could be considered as a potentially precise and low detection error method for further large amounts of inspections in a real environment.

A comparative study on optimization methods for experiments with ordered categorical data

When conducting experiments, the selected quality characteristic should as far as possible be a continuous variable and be easy to measure. Due to the inherent nature of the quality characteristic or the convenience of the measurement technique and cost-effectiveness, the data observed in many experiments are ordered categorical. To analyze ordered categorical data for optimizing factor settings, there are three widely accepted approaches: Taguchis accumulation analysis, Nairs scoring scheme and Jengs weighted probability scoring scheme. In this paper, a simpler method named the weighted SN ratio method for analyzing ordered categorical data is introduced. A case study involving optimizing the polysilicon deposition process for minimizing surface defects and achieving the target thickness in a very large-scale integrated circuit can demonstrate the four approaches. Finally, comparative analyses of efficiency for employing the four approaches to optimize factor settings are presented according to simulated experimental data that are normally, Weibull and Gamma distributed. From the results, it is obvious that the weighted SN ratio method has the properties of easy computation and uses one-step optimization to obtain the optimal factor settings. Its efficiency is slightly less than that of the scoring scheme, better than that of the accumulation analysis and the weighted probability-scoring scheme.

Sequential optimization of parameter and tolerance design for multiple dynamic quality characteristics

System design, parameter design and tolerance design are the three stages of design process as presented by G. Taguchi. Systems design identifies the basic elements of the design to provide new or improved products to customers. Parameter design determines the optimal parameter settings, which will minimize variation from the target performance of the product. Tolerance design finally identifies the components of the design, which are sensitive in terms of affecting the quality of the product, and establishes tolerance limits that will give the required level of variation in the design. Most studies have focused primarily on optimizing the parameter design or tolerance design for multiple static quality characteristics. In this paper, a mathematical formula corresponding to the model is derived from Taguchis quadratic quality loss function to minimize the expected total cost for the parameter design of multiple dynamic quality characteristics. When the optimal parameter design is not sufficient to reduce the output variation, the first-order Taylor series expansion is then used to analyse the variations of noise factors for optimizing the tolerance design. It concludes with an example demonstrating this approach.

Validation and evaluation for defect-kill-rate and yield estimation models in semiconductor manufacturing

This study evaluates the accuracy of two alternative models for the kill rate of different visual defects and the yield estimation by using large amount of practical inspection data (140 6-inch wafers containing 70 560 dies) in semiconductor manufacturing. One model assumed that the visual defects are randomly distributed on a wafer die; another model considered that clustering visual defects may occur on a wafer die. The results show that two models are both capable of predicting the yield precisely. The results also suggest that the model assuming clustering visual defects on a die is more accurate based on the analysis of Taguchis signal-to-noise (SN) ratio. The same practices can be implemented with other types of wafer such as 8 inch and 12 inch. Precise prediction for the kill rate of different types of visual defects and the yield in the long supply of semiconductor industry is critical since the requirement of material and the reserved capacity can be largely reduced.

Robust design of nonlinear dynamic problem

Taguchis robust design is a popular methodology utilized within many different industries in order to improve product quality and process performance at low cost. The robust design of dynamic system is used to find the optimal control factor settings so that the measured response is closest to the ideal function. Although Taguchis robust design was effective in improving performance of products or processes, the statisticians pointed to inefficiencies in the method for highly nonlinear problems and complexity of the parameter design. This paper proposes a general model for optimizing the robust design of dynamic system parameter based on the quality loss function. The objective is to minimize the average quality loss instead of maximizing SN ratio suggested by Taguchi. The example of high-precision positioning device is provided to demonstrate the implementation and effectiveness of the proposed method.

Boundary-based passive component inspection approach using eigenvalues of covariance matrices

A boundary-based scheme to inspect basic dimensions and surface defects on solder plates (terminations) of passive components by using the eigenvalues of covariance matrices was developed. The image of the passive component was initially processed to show only two terminations. The eigenvalues of covariance matrices for boundary points, which are equivalent to curvature measurements, then generate a one-dimensional representation describing the angle variations on termination boundaries. Since a single corner (intersection of two boundary edges) and jagged corners (a surface defect) are local deviations on termination boundaries, their locations will be represented as highly fluctuating eigenvalue waveforms. By setting an appropriate threshold on the eigenvalues, the surface defects and single corners on the termination boundaries can be identified. The basic dimensions of a passive component are determined simply by the distances between the detected single corners. Real passive components (0805 MLCCs) are testing samples to evaluate the performance of the proposed approach. Experimental results show that the proposed approach achieves precise identification for surface defects and measurement for the basic dimensions. The proposed approach is precise, rotation invariant and template free. Therefore, it is especially suitable for small batch production of various types of passive components.

ROBUST DESIGN OF DIGITAL-DIGITAL DYNAMIC SYSTEM

Problems where the output varies depending on the input are classified as dynamic systems in the Taguchi method. In dynamic system, if both input and output have only two digital values (0 and 1) with the possibility of committing two types of errors (judging 0 as 1 and 1 as 0), such a problem is called ”digital-digital” dynamic problem. Taguchi recommended calculating the equalized error rates first and then obtaining the standardized signal-to-noise ratio based on the two errors with the same loss coefficient to optimize the parameter settings for such a problem. However, the losses due to the two types of errors are not equal in practice, especially in the field of chemistry. This paper proposes a general model for analyzing the digital-digital dynamic system. The implementation and the effectiveness of the proposed approach are illustrated through two cases.