隨著晶片多功能及輕薄趨勢發展，晶片與晶片線路間隔愈來愈密集，然因對於晶片堆疊所需之間隔高密度需求，其初期的封裝銅柱凸塊穩定性不一，尤以銅柱凸塊在長成時的成型直徑為對於晶片效能以及功率是主要的管控項目。在新產品開發初期，囿於製程穩定性以及試製產品樣本稀少，統計製程管制圖並無法提供予工程師足夠之資訊，因此本研究結合盒鬚圖與模糊理論，開發出一個考量資料動態趨勢之灰預測模式，藉以改善傳統灰預測GM(1,1)模式對短期時間序列資料的預測績效。實驗結果顯示，本研究所提出之灰預測模式除增進對個案資料之精準度外，亦在公開測試資料中較其他多種灰預測模式有顯著更佳之預測準確度。

In new product build up, SPC data that gather by insufficient prior run samples is not complete enough to offer production performance for engineering’s study and evaluation,especially for semi-conductor technology progress.
As multiple and thin function chip development,fine pitch design we called copper pillar is capable to achieve high density layout requirement.By this solution,bump diameter that may influence chip function and efficiency takes the most important role to fulfill this technology development. In order to evaluate and simulate production performance, we are using box plot and fuzzy theory that consider dynamic tendency and grey model for learning short-term time series production data,to improve the short term time series prediction performance of traditional Gray Model (1,1).
According to simulation result,this dynamic tendency grey model provided the best prediction accuracy for the research case and the other public testing data,than the rest gray model.

Alcock, R. J., Manolopoulos, Y, Time-Series Similarity Queries Employing a Feature-Based Approach. 7th Hellenic Conference on Informatics, Ioannina, Greece, 1-9, 1999.
Chen, C. I., Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate. Chaos, Solitons and Fractals, 37(1), 278-287, 2008.
Chen, C. I., Chen, H. L., Chen, S. P., Forecasting of foreign exchange rates of Taiwan’s major trading partners by novel nonlinear grey Bernoulli model NGBM(1,1). Communications in Nonlinear Science and Numerical Simulation, 13(6), 1194-1204, 2008.
Chen, H. S., Chang, W. C., A study of optimal grey model GM(1,1). Journal of Grey System, 1(2), 141-145, 1998. (in Chinese)
Chen, K. W., Lai, C. J., Optimal fixed α in for GM(1,1). The 6th National Conference on Grey Theory and Applications, Yunlin, Taiwan, A26-A31, 2001. (in Chinese)
Cheng, K. H., Shah, H. C., A new method for earthquake forecasting using gery theory: Application to California. The Journal of Grey System, 11(3), 293-302, 1999.
Chang, S. C., Lai, H. C., Yu, H. C., A variable P value rolling grey forecasting model for Taiwan semiconductor industry production. Technological Forecasting & Social Change, 72(5), 623-640, 2005.
Chang, S. C., Wu, J. H., Lee, C. T., A study on the characteristics of α(k) of grey prediction. The 4th National Conference on Grey Theory and Applications, Kaohsiung, Taiwan, 291-296, 1999. (in Chinese)
Deng, J. L., Control Problems of Grey Systems. Systems and Control Letters, 1(5), 288-294, 1982.
Deng, J. L., Grey System Fundamental Method. Huazhong University of Science and Technology Press, Wuhan, China, 1987. (in Chinese)
Deng, J. L., Introduction to Grey System Theory. The Journal of Grey System, 1(1), 1-24, 1989.
El-Fouly, T. H. M., El-Saadany, E. F., Salama, M. M. A., Grey predictor for wind energy conversion systems output power prediction. IEEE Transactions on Power Systems, 21(3), 1450-1452, 2006.
He, Y., Bao, Y. D., Grey-Markov forecasting model and its application. Systems Engineer-Theory & Practice,12(4), 59-63, 1992. (in Chinese)
Hsin, J. Y., Tsai, Y. P., The research of superposition method for α value in grey forecasting. The 5th National Conference on Grey Theory and Applications, Taipei, Taiwan, 305-308, 2000. (in Chinese)
Hsu, C. C., Chen, C. Y., Applications of improved grey prediction model for power demand forecasting. Energy Conversion and Management, 44(14), 2241-2249, 2003a.
Hsu, C. C., Chen, C. Y., A modified Grey forecasting model for long-term prediction. Journal of the Chinese Institute of Engineers, 26(3), 301-308, 2003b.
Hsu, C. I., Wen, Y. H., Applying grey forecasting models to predict international air travel demand for Taiwan area. Transportation Planning Journal, 26(3), 525-556, 1997. (in Chinese)
Hsu, C. I., Wen, Y. H., Improved grey prediction models for the trans-pacific air passenger market. Transportation Planning and Technology, 22(2), 87-107, 1998.
Hsu, L. C., Applying the Grey prediction model to the global integrated ciruit industry. Technological Forecasting & Social Change, 70(6), 563-574, 2003.
Huang, C., Principle of information diffusion. Fuzzy Sets and Systems, 91(1), 69-90, 1997.
Huang, C., Moraga, C., A diffusion-neural-network for learning from small samples. International Journal of Approximate Reasoning, 35(2), 137-161, 2004.
Hung, M., He, Y., Cen, H., Predictive analysis on electric-poewr supply and demand in China. Renewable Energy, 32(7), 1165-1174, 2007.
Jang, J. S. R., ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man and Cybernetics, 23(3), 665-685, 1993.
Liang, E. B., Application of grey system theory to the forecast of China’s steel output. Iron and Steel, 24(11), 70-73, 1989. (in Chinese)
Li, D. C., Chang, C. J., Chen, C. C., & Chen, C. S. (2010). Using Non-equigap Grey Model for Small Data Set Forecasting - A Color Filter Manufacturing Example. Journal of Grey System, 22(4), 375-382.
Li, D. C., Chen, C. C., Chang, C. J., & Chen, W. C. (2011). Employing Box-and-Whisker plots for learning more knowledge in TFT-LCD pilot runs. International Journal of Production Research (In Press).
Li, D. C., Chen, L. S., Lin Y. S., Using functional virtual population as assistance to learn scheduling knowledge in dynamic manufacturing environments. International Journal of Production Research, 41(17), 4011-4024, 2003.
Li, D. C., & Yeh, C. W. (2008). A non-parametric learning algorithm for small manufacturing data sets. Expert Systems with Applications, 34(1), 391-398.
Li, D. C., Lin, Y. S., Using virtual sample generation to build up management knowledge in the early manufacturing stage. European Journal of Operational Research, 175(1), 413-434, 2006a.
Li, D. C., Wu, C. S., Chang, F. M., Using data-fuzzification technology in small data set learning to improve FMS scheduling accuracy. International Journal of Advanced Manufacturing Technology, 27(3-4), 321-328, 2005.
Li, D. C., Wu, C. S., Tsai, T. I., Chang, F. M., Using mega-fuzzification and data trend estimation in small data set learning for early FMS scheduling knowledge. Computer & Operations Research, 33(6), 1857-1869, 2006b.
Li, D. C., Wu, C. S., Tsai, T. I., Lina, Y. S., Using mega-trend-diffusion and artifical samples in small data set learning for early flexible manufacturing system scheduling knowledge. Computer & Operations Research, 34(4), 966-982, 2007.
Li, D. C., Yeh C. W., A non-parametric learning algorithm for small manufacturing data sets. Expert Systems with Applications, 34(1), 391-398, 2008.
Li, Y. G., Li, Q. F., Zhao, G. F., An improvement on Grey forecasting model. System Engineering, 10(6), 27-31, 1992. (in Chinese)
Lin, Y. S., & Li, D. C. (2010). The Generalized-Trend-Diffusion modeling algorithm for small data sets in the early stages of manufacturing systems. European Journal of Operational Research, 207(1), 121-130.
Lin, C. T., Yang, S. Y., Forecast of the output value of Taiwan’s opto-electronics industry using the Grey forecasting model. Technological Forecasting & Social Change, 70(2), 177-186, 2003.
Lin, C. T., Yeh, H. Y., The use of grey prediction to forecast Taiwan stock index option prices. The Journal of Grey System, 18(4), 381-390, 2006.
Lin, Y. S., An incremental learning algorithm from small sequential manufacturing data sets. Doctoral dissertation, National Cheng Kung University, Tainan, Taiwan, 2006.
Liu, S. F., Dang, Y. G., Fang, Z. G., The Theory of Grey System and Its Applications. Science Press, Beijing, China, 2004. (in Chinese)
Liu, W. G., Jiang, L. H., Grey GM(1,1) model in ferroalloy burdening. Iron and Steel, 31(9), 52-56, 1996. (in Chinese)
Lu, H. C., Universal GM(1,1) model based on data mapping concept. The Journal of Grey System, 8(4), 307-319, 1996.
Luo, D., Liu, S. F., Dang, Y. G., The optimization of grey model GM(1,1). Engineering Science, 5(8), 50-53, 2003. (in Chinese)
Luo, E. X., Qian, X. S., Li, R., Construction and empirical research of the variable parameter value rolling grey forecasting model. Journal of University of Shanghai for Science and Technology, 28(5), 465-468, 2006. (in Chinese)
Man, L., An application of GM(1,1) model: The prediction og flight safety. The journal of Grey System, 1(1), 99-102,1989.
Mao, M., Chirwa, E. C., Combination of grey model GM(1,1) with three-point moving average for accurate vehicle fatality risk prediction. International Journal of Crashworthiness, 10(6), 635-642, 2005.
Mao, M., Chirwa, E. C., Application of grey model GM(1,1) to vehicle fatality risk estimation. Technological Forecasting & Social Change, 73(5), 588-605, 2006.
Niyogi, P., Girosi, F., Poggin, T., Incorporating prior information in machine learning by creating virtual examples. Proceedings of the IEEE, 86(11), 2196-2209, 1998.
Pan, C. L., Huang, Y. F., Lin, G., The study on α algorithm of grey prediction with iterative method as basis. The 7th National Conference on Grey Theory and Applications, Tainan, Taiwan, I27-I32, 2002.
Song, Z. M., Tong, X. J. Xiao, X. P., Center approach grey GM(1,1) model. Systems Engineer-Theory & Practice, 21(5), 110-113, 2001. (in Chinese)
Sun, G., Prediction of vegetable yields by grey model GM(1,1). The Journal of Grey System, 3(2), 179-187, 1991.
Tan, G.. J., The structure method and application of background value in grey system GM(1,1) model (Ⅰ). Systems Engineer-Theory & Practice, 20(4), 98-103, 2000. (in Chinese)
Tien, T. L., Chen, S.P, Residual correction method of Fourier series to GM(1,1) Model. The 1st National Conference on Grey Theory and Applications, Kaohsiung, Taiwan, 93-101, 1996. (in Chinese)
Tukey, J. W. (1977). Exploratory data analysis: Reading (MA): Addison-Wesley.
Wang, Y. F. (2003). On-demand forecasting of stock prices using a real-time predictor. IEEE Transactions on Knowledge and Data Engineering, 15(4), 1033-1037.
Wang, Y., Song, Q. B., MacDonell, S., Shepperd, M., & Shen, J. Y. (2009). Integrate the GM(1,1) and Verhulst Models to Predict Software Stage Effort. IEEE Transactions on Systems Man and Cybernetics Part C-Applications and Reviews, 39(6), 647-658.
Wen, K. L., Huang, Y. F., Chen, F. S., Lee, Y. B., Lian, Z. F., Lai, J. R., Grey Prediction. Chuan Hwa Book Press, Taipei, Taiwan, 2002. (in Chinese)
Xiao, X. P., & Li, F. Q. (2009). Research on the stability of non-equigap grey control model under multiple transformations. Kybernetes, 38(10), 1701-1708.
Yao, A. W. L., Chi, S. C., Chen, J. H., An improved Grey-based approach for electricity demand forecasting. Electric Power Systems Research, 67(3), 217-224, 2003.
Yokum, J. T., Armstrong, J. S., Beyond accuracy: Comparison of criteria used to select forecasting methods. International Journal of Forecasting, 11(4), 591-597, 1995.
Zhang, H., Li, Z., Chen, Z., Application of grey modeling method to fitting and forecasting wear trend of marine diesel enfines. Tribology International, 36(10), 753-756, 2003.
Zhou, C. Y., Li, D. F., Liu, Z. X., Grey Predicting the soft ground settlement via GM(1,1). The Journal of Grey System, 11(4), 397-402, 1999.
Zhou, P., Ang, B. W., Poh K. L., A trigonometric grey prediction approach to forecasting electricity demand. Energy, 31(14), 2839-2847, 2006.