在製造環境迅速變遷的今日，如何有效地控制製造系統已成為企業必須面對的重要議題。尤其，製造系統的早期階段更是管理的關鍵，在僅能獲得有限資訊的這個時期，如何做出正確決策往往是管理者必須克服的難題。為此，本研究發展一個兩階段的模型建構程序，以做為小樣本資料的預測工具。本研究先利用線性迴歸、倒傳遞神經網路與支撐向量迴歸等三種常見的方法建立模型，之後利用灰色配適衡量指標進行預檢驗並確定模型的權重，最後使用加權平均的方式產生預測模型。經過彩色濾光片製造資料與亞太經濟合作會議能源資料庫的實際測試，本研究方法能有效降低預測誤差的變異性，且平均絕對百分誤差分別為2.076%與2.670%，擁有良好的預測效果，顯見本研究提出的建模程序是一個處理小樣本預測的可行方法。

Enterprises need to achieve effective and efficient control of production systems at an early stage of setting up a new manufacturing process, when systems are newly built, and decisions that are made at this point are generally based on insufficient information. To address this issue, the paper develops a two-stage procedure to build a predictive model using few samples. In the first stage, three learning approaches, including linear regression, a back propagation neural network, and support vector regression, are employed to establish forecasting models; while in the second stage, a grey-based fitness measuring index is proposed to implement pre-testing to determine the weights of these models to create a hybrid model. Two datasets, including color filter manufacturing data and the Asia-Pacific Economic Cooperation energy database, are evaluated in the experiment, and the results show that the proposed method can lower the variance of forecasting errors and achieve good performance (where the mean absolute percentage errors in the two datasets are 2.076% and 2.670%, respectively). Accordingly, the proposed procedure is thus considered a feasible approach for small-data-set forecasting.

Abu-Mostafa, Y. S., Learning from hints in neural networks. Journal of Complexity, 6(2), 192-198, 1990.
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.
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.
Cheng, K. H., Shah, H. C., A new method for earthquake forecasting using gery theory: Application to California. Journal of Grey System, 11(3), 293-302, 1999.
Cho, S. Z., Jang, M., Chang, S. J., Virtual sample generation using a population of networks. Neural Processing Letters, 5(2), 83-89, 1997.
Chui, F., Elkamel, A., Surit, R., Croiset, E., Douglas, P. L., Long-term electricity demand forecasting for power system planning using economic, demographic and climatic variables. European Journal of Industrial Engineering, 3(3), 277-304, 2009.
Dalrymple, D. J., Sales forecasting practices: Results from a United States survey. International Journal of Forecasting, 3(3-4), 379-391, 1987.
Deng, J. L., Control Problems of Grey Systems. Systems and Control Letters, 1(5), 288-294, 1982.
Deng, J. L., Introduction to Grey System Theory. Journal of Grey System, 1(1), 1-24, 1989.
Ediger, V. S., Akar, S, ARIMA forecasting of primary energy demand by fuel in Turkey. Energy Policy, 35(3), 1701-1708, 2007.
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.
Hamzacebi, C., Forecasting of Turkey's net electricity energy consumption on sectoral bases. Energy Policy, 35(3), 2009-2016, 2007.
Hsu, L. C., Applying the Grey prediction model to the global integrated circuit industry. Technological Forecasting & Social Change, 70(6), 563-574, 2003.
Hsu, L. C., A genetic algorithm based nonlinear grey Bernoulli model for output forecasting in integrated circuit industry. Expert Systems with Applications, 37(6), 4318-4323, 2010.
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., Improved grey prediction models for the trans-pacific air passenger market. Transportation Planning and Technology, 22(2), 87-107, 1998.
Hu, M. Y., Zhang, G. Q., Jiang, C. Z., Patuwo, B. E., A cross-validation analysis of neural network out-of-sample performance in exchange rate forecasting. Decision Sciences, 30(1), 197-216, 1999.
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.
Ivanescu, V. C., Bertrand, J. W. M., Fransoo, J. C., Kleijnen, J. P. C., Bootstrapping to solve the limited data problem in production control: an application in batch process industries. Journal of the Operational Research Society, 57(1), 2-9, 2006.
Jang, J. S. R., ANFIS: Adaptive-network-based fuzzy inference system. IEEE Transactions on Systems, Man and Cybernetics, 23(3), 665-685, 1993.
Lee, Y. S., Tong, L. I., Forecasting energy consumption using a grey model improved by incorporating genetic programming. Energy Conversion and Management, 52(1), 147-152, 2011.
Li, D. C., Chang, C. J., Chen, W. C., Chen, C. C., An extended grey forecasting model for omnidirectional forecasting considering data gap difference. Applied Mathematical Modelling, 35(10), 5051-5058, 2011.
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., Fang, Y. H., A non-linearly virtual sample generation technique using group discovery and parametric equations of hypersphere. Expert Systems with Applications, 36(1), 844-851, 2009.
Li, D. C., Fang, Y. F., Lai, Y. Y., Hu, S. C. Utilization of virtual samples to facilitate cancer identification for DNA microarray data in the early stages of an investigation. Information Sciences, 179(16), 2740-2753, 2009a.
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, 2006.
Li, D. C., Lin, Y. S., 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, 2010.
Li, D. C., Tsai, T. I., Shi, S., A prediction of the dielectric constant of multi-layer ceramic capacitors using the mega-trend-diffusion technique in powder pilot runs: case study. International Journal of Production Research, 47(1), 51-69, 2009b.
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, 2006.
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, D. C., Yeh, C. W., Chang, C. J., An improved Grey-based approach for early manufacturing data forecasting. Computers and Industrial Engineering, 57(4), 1161-1167, 2009c.
Li, D. C., Yeh, C. W., Li, Z. Y., A case study: The prediction of Taiwan’s export of polyester fiber using small-data-set learning methods. Expert Systems with Applications, 34(3), 1983-1994, 2008.
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. Journal of Grey System, 18(4), 381-390, 2006.
Liu, S. F., Lin, Y., Grey Information: theory and practical applications. 1st ed., Springer, London, Britain, 2006.
Mahmoud, E., Rice, G., Malhotra, N., Emerging issues in sales forecasting and decision support systems. Journal of the Academy of Marketing Science, 16(3-4), 47-61, 1988.
Man, L., An application of GM(1,1) model: The prediction of flight safety. 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.
Pao, H. T., Forecasting electricity market pricing using artificial neural networks. Energy Conversion and Management, 48(3), 907-912, 2007.
Pao, H. T., Forecasting energy consumption in Taiwan using hybrid nonlinear models. Energy, 34(10), 1438-1446, 2009.
Sun, G., Prediction of vegetable yields by grey model GM(1,1). Journal of Grey System, 3(2), 179-187, 1991.
Tsai, T. I., Li, D. C., Utilize bootstrap in small data set learning for pilot run modeling of manufacturing systems. Expert Systems with Applications, 35(3), 1293-1300, 2008.
Wen, Q. Y., Zhang, H. W., Zhang, P. X., Jiang, X. D., Improved artificial neural network for data analysis and property prediction in slag class-ceramic. Journal of the American Ceramic Society, 88(7), 1765-1769, 2005.
Witten, I. H., Frank, E., Data mining: practical machine learning tools and techniques. 2nd ed., Morgan Kaufmann Publishers, San Francisco, United States, 2005.
Yang, J., Yu, X., Xie, Z. Q., Zhang, J. P., A novel virtual sample generation method based on Gaussian distribution. Knowledge-Based Systems, 24(6), 740-748, 2011.
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, P., Ang, B. W., Poh, K. L., A trigonometric grey prediction approach to forecasting electricity demand. Energy, 31(14), 2839-2847, 2006.
Zhou, C. Y., Li, D. F., Liu, Z. X., Grey Predicting the soft ground settlement via GM(1,1). Journal of Grey System, 11(4), 397-402, 1999.