公差設計為機械設計與製造中之重要環節，常被用來權衡品質和成本之間的取捨，適當訂定公差精度可以提升產品品質且節省生產成本，若要進行公差設計就需要取得生產活動中大量產品偏差值之數據進行統計分析。拜電腦科技優化統技法進步之賜，現代工業產品得以透過電腦技術進行生產環境、產品規格，與各項設計製造程序之系統性整體優化。
本文以多體系統分析為出發點，透過多體系統分析推導出組合件之約束方程式並且得到組合件中的關鍵節點，將節點設定公差精度後再利用蒙地卡羅法模擬零件之生產，最後透過優化法計算得到模擬組合件之誤差與其相對總生產成本(comprehensive cost)再進一步比較不同精度設定下之相對總生產成本，其中最低相對總生產成本之精度設定即為最佳公差精度設定。本文中使用了三種模型，即四孔模型、四聯桿模型和實際工具機煞車機構模型作為範例，實際運用本文所呈之研究方法進行分析、模擬，以得出數據與分析結果結果後並與文獻、廠商實際訂定之公差精度設定下生產產品準確度和廠商常用之傳統公差設計法作比較，視其結果能否顯示本文所用之研究方法相較於其他研究方法的獨特之處。
為了得到機械系統中各個零件之最佳公差精度設定，本文透過四孔模型、四聯桿模型和工具機煞車系統模型使用公差設計優化法計算在可能的生產製造精度範圍下所有精度設定之排列組合，找出最佳的公差精度設定。從公差設計優化法模擬實驗得到的結果可以歸納為以下五點結論，首先將公差設計優化法利用旅行商人問題引入，並應用於解決不同形態組合件之公差設計問題。其次，由模擬零件生產之製程能力指數得知，公差設計優化法之模擬數據較為準確，模擬四孔模型的結果顯示公差設計優化法可應用於靜態系統，接著利用四聯桿模型之模擬結果實證公差設計優化法亦可應用於三維動態系統，最後透過真實的機械煞車系統模擬，將公差設計優化法應用於現實業界中實際生產之產品中，優化後之最佳公差精度設定符合ISO之規範，代表著公差設計優化法不僅能夠優化假想的組合件或機構更可以應用於實際生產之產品，滿足設計者決定各個零件公差之需求

Tolerance allocation is a significant procedure in mechanical design and production, it is a bridge that links product performance and requirements, and relate closely with product quality and cost. Allocating proper tolerance can improve product quality and save manufacturing cost. However, allocating tolerance needs lots of data of product manufacturing to analysis. Thanks to the improvement of computing technology, the environment of manufacturing, specification and designing procedure of industrial products are systematic optimized through computing technology.

In order to obtain the best tolerance allocation of each components in mechanical system, we use three kinds of model, which are four holes model, four-bar linkage model and brake system in mechanism model, to calculate and find the best tolerance allocation of all combinations in investigate range by tolerance optimal method. The simulations show that tolerance optimal method can apply in the static system, the dynamic system in 3D and brake system of mechanism system. In addition, the results obtained by tolerance optimal method are more accurate than other methods. It prove that tolerance method can optimal the tolerance allocation of assembly or mechanism. Moreover, it can apply in the manufacturing of reality products, fulfill the needs of tolerance allocation problem from designer.

[1] Drake, P. J., “Dimensioning and Tolerancing Handbook,” 1999.
[2] Chase, K.W., “Tolerance Allocation Methods for Designers,” Brigham Young University, 1999.
[3] Choi, H. G. R., Park, M. H. and Salisbury, E., “Optimal tolerance allocation with loss functions,” Journal of Manufacturing Science and Engineering, 122(3), 529-535, 2000.
[4] Ji, S., Li, X., Ma, Y., and Cai, H., “Optimal Tolerance Allocation Based on Fuzzy Comprehensive Evaluation and Genetic Algorithm,” Harbin Institute of Technology, 2000.
[5] Ertuğrul, I. and Güneş, M., “The Usage of Fuzzy Quality Control Charts to Evaluate Product Quality and an Application,” University of Pamukkale, 2007.
[6] Lai, Y.S., “Truss-structure Optimization using Ant Algorithm,” Tatung University, 2004.
[7]Li, S.Y., “Ant Colony Algorithms with Application,” Harbin Institute of Tchnology, 2004.
[8] Hoffenson, S., Dagman, A. and Söderberg, R., “ A Multi-objective Tolerance Optimization Approach for Economic, Ecological, and Social Sustainability,” Chalmers University of Technology, 2013.
[9] Mazur, M., “Tolerance analysis and synthesis of assemblies subject to loading with process integration and design optimization tools,” RMIT University Melbourne, 2013.
[10] Jayaprakash, G. and Sivakumar, K., “Parametric Tolerance Analysis of Mechanical Assembly Using FEA and Cost Competent Tolerance Synthesis Using Neural Network,” Shivani Engineering College, 2010.
[11] Yeo, S.H., Ngoi, B.K.A., Poh, L.S. and Hang, C., “Cost-Tolerance Relationships for Non-Traditional Machining Processes,” The international journal of advanced manufacturing technology, pp. 35-41, 1997
[12] Chun, H., Kwon, S.J. and Tak, T., “Multibody approach for tolerance and optimization of mechanical systems,” Journal of Mechanical Science and Technology, pp. 276-286, 2008.
[13] Mao, J., Li B., Zhang, H., and Cao, Y.l., “An experimental design method for robust tolerance design,” Shanghai University of Engineering Science.
[14] Cao, Y.L., Mao, J., Yang, J.X., Wu, Z.T. and Wu, L.Q., “A robust tolerance design method based on fuzzy quality loss,” Frontiers of Mechanical Engineering in China, pp. 101-105, 2006.
[15] Tichkiewitch, S., Tollenaere, M. and Ray, P., “Advances in Integrated Design and Manufacturing in Mechanical Engineering II,” Springer, 2007.
[16] Agrawal, A.A., Strauss, S.Y. and Stout, M.J., “Cost of induced responses and tolerance to herbivory in male and female fitness components of wild radish,” Evolution, pp. 1093-1104, 1999.
[17] Dimitrellou, S.C., Diplaris, S.C. and Sfantsikopoulos, M.M., “A systematic approach for cost optimal tolerance design,” In Proceedings of the International Conference On Engineering Design-ICED’07, pp. 28-31, 2007.
[18] Kim, K. J., Moskowitz, H., Dhingra, A. and Evans, G., “Fuzzy multicriteria models for quality function deployment,” European Journal of Operational Research, pp. 504-518, 2000.
[19] Jayaprakash, G. and Sivakumar, K., “Parametric tolerance analysis of mechanical assembly using FEA and cost competent tolerance synthesis using neural network,” Journal of Software Engineering and Applications, pp. 1148, 2010.
[20] Bernardo, F.P. and Saraiva, P.M., “Robust optimization framework for process parameter and tolerance design,” University of Coimbra, 1998.
[21] Cagan, J. and Kurfess, T.R., “Optimal design for tolerance and manufacturing allocation,” Carnegie Mellon University, 1991.
[22] Pan, J.N. and Pan, J., “Optimization of engineering tolerance design using revised loss functions,” Engineering optimization, pp. 99-118, 2009.
[23] Maier, H.R., Simpson, A.R., Zecchin, A.C., Foong, W.K., Phang, K.Y., Seah, H.Y. and Tan, C.L., “Ant colony optimization for design of water distribution systems,” Journal of water resources planning and management, pp. 200-209, 2003.
[24] Chen, Y.A., and Huang, K.C., “An ant-based algorithm for the dynamic berth allocation problem,” National Chiao Tung University.
[25] Socha, K. and Dorigo, M., “Ant colony optimization for continuous domains,” European journal of operational research, pp. 1155-1173, 2008.
[26] McMullen, P.R., “An ant colony optimization approach to addressing a JIT sequencing problem with multiple objectives,” Artificial Intelligence in Engineering, pp. 309-317, 2001.
[27] Parpinelli, R.S., Lopes, H.S. and Freitas, A., “Data mining with an ant colony optimization algorithm,” Evolutionary Computation, IEEE Transactions on, pp. 321-332, 2002.
[28] Merkle, D., Middendorf, M. and Schmeck, H., “Ant colony optimization for resource-constrained project scheduling,” Evolutionary Computation, IEEE Transactions on, pp. 333-346, 2002.
[29] Sim, K.M. and Sun, W.H., “Ant colony optimization for routing and load-balancing:survey and new directions,” Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, pp. 560-572, 2003.
[30] Liang, Y.C. and Smith, A.E., “An ant colony optimization algorithm for the redundancy allocation problem (RAP),” Reliability, IEEE Transactions on, pp. 417-423, 2004.
[31] Obayashi, S., “Multi-objective design exploration using efficient global optimization,” European Community on Computational Methods in Applied Sciences (ECCOMAS), 2006.
[32] Jin, S., Zheng, C., Yu, K. and Lai, X., “Tolerance design optimization on cost-quality trade-off using the Shapley value method,” Journal of Manufacturing Systems, pp. 142-150, 2011.
[33] Andolfatto, L., Thiébaut, F., Lartigue, C. and Douilly, M., “Quality- and cost –driven assembly technique selection and geometrical tolerance allocation for mechanical structure assembly,” Journal of Manufacturing Systems, pp. 103-115, 2014.
[34] Milberg, C., Tommelein, I.D. and Alves, T., “Improving design fitness through tolerance analysis and tolerance allocation,” University of California, Berkeley, 2002.
[35] Hung, T.C. and Chan, K.Y., “Multi-objective design and tolerance allocation for single and multi-level systems,” Journal of Intelligent Manufacturing, pp. 559-573, 2011.
[36] Altarazi, S.A., “Operational tolerance allocation and machine assignment under process capability and product value constraints,” Wichita State University, 2000.
[37] “製程能力指數 Cpk,” Northwest Analytics. Retrieved from http://taiwan.vulcan-tech.com/cpk/.
[38] “過程能力指數,” MBA智庫百科. Retrieved from http://wiki.mbalib.com/zhtw/%E8%BF%87%E7%A8%8B%E8%83%BD%E5%8A%9B%E6%8C%87%E6%95%B0.