隨著產品生命週期縮短，產線需要不斷進行調整，為了能確保製造系統的產出能滿足顧客的需求，我們需要一個強而有力的指標。在許多產業已經運用網路可靠度作為一項重要評估製造系統是否穩健的項目，目前相關的可靠度計算方法，主要是以精確求法為主。然而，精確求法的核心在於列舉所有的可能狀況，當加工機台數量增加，或是在產線上具有多次重工的情況，或是輸入量與需求量之間的差異變大時，精確求法的計算複雜度會呈現指數增長，因隨著製造網路的複雜化，在可行解時需要花費非常可觀的時間，因此當我們面對大型製造網路時，可靠度的計算顯得耗時費力。
為此，本研究提出了一種快速且精確的估計算法，所提之演算法是以蒙地卡羅模擬法為模擬基礎，考慮各工作站的產能以及機台加工的成功率，計算產線的最終產出，並依據產出的結果是否能滿足顧客需求，來估算網路可靠度。為了驗證我們模型的有效性，本研究列舉了四個範例，面對簡易的網路，我們以桑教授 [1] 所提出的Song rule作為精確值，近似算法作為預測值，面對較複雜的網路，因精確值計算需要耗時相當久，我們以桑教授所提的模擬算法來取代Song rule 作為精確值，並且計算MAPE來評估網路可靠度的預測誤差屬何種類型的精確度。從四個範例中，我們得知隨著製造網路的複雜化，在維持高度精確度下，近似算法與模擬算法的計算時間差距會越加明顯。我們提出的方法不僅可以適用於各型態的網路，且因為是一種簡單、有效且快速的方法，非常適合產線人員用來計算製造現場的網路可靠度。

With the shortening of the product life cycle, the production line needs to be constantly adjusted. In order to ensure that the output of the manufacturing system can meet the needs of customers, we need a strong indicator. Network reliability has been used as an important project to evaluate the robustness of manufacturing systems in many industries. However, accurate calculation methods of the core is to list all the possible conditions, when processing the machine number, or on the production line has many heavy industries, or the difference between input and demand grows, the accurate calculation methods of computational complexity can present exponential growth, because along with the manufacturing of complicated network, when feasible solution need to spend a very considerable time, so when we face big manufacturing network, the calculation of reliability is time-consuming.
For this reason, a fast and accurate estimation algorithm is proposed in this study. Based on the monte carlo simulation method, the final output of the production line is calculated by considering the productivity of each workstation and the success rate of machine processing. In order to demonstrate the effectiveness of our model, this study enumerates four examples, in the face of simple network, we put forward to mulberry professor Song rule as a precise value, approximate algorithm as predicted, in the face of complex network, due to the need of precise terms take quite a long time, we sang, a professor at the proposed simulation algorithm to replace Song rule as a precise value, and calculate the MAPE to evaluate network reliability prediction error is what type of accuracy. From the four examples, we know that with the complexity of the manufacturing network, the computational time difference between the approximate algorithm and the simulation algorithm will become more and more obvious while maintaining a high degree of accuracy. Our proposed method is not only applicable to all types of networks, but also is a simple, effective and fast method, which is very suitable for manufacturing personnel to calculate the network reliability of manufacturing site.

[1] W. T. Song, "The Song rule as a validator of analytical results—A note correcting system reliability results in a review of the literature", IEEE Transactions on Reliability, vol. 66, no. 4, pp. 1012-1024, 2017.
[2] A. Adamyan and D. He, "Analysis of sequential failures for assessment of reliability and safety of manufacturing systems", Reliability Engineering System Safety, vol. 76, no. 3, pp. 227-236, 2002.
[3] Y. Liu and J. Li, "Modelling and analysis of split and merge production systems with Bernoulli reliability machines", International Journal of Production Research, vol. 47, no. 16, pp. 4373-4397, 2009.
[4] W. J. Stevenson, M. Hojati, and J. Cao, "Operations management", McGraw-Hill/Irwin Boston, 2007.
[5] T. Aven, "Availability evaluation of oil/gas production and transportation systems", Reliability engineering, vol. 18, no. 1, pp. 35-44, 1987.
[6] T. Aven, "Some considerations on reliability theory and its applications", Reliability Engineering System Safety, vol. 21, no. 3, pp. 215-223, 1988.
[7] F. S. Roberts, F. Hwang, and C. L. Monma, "Reliability of computer and communication networks: proceedings of a DIMACS workshop, December 2-4, 1989", American Mathematical Soc., 1991.
[8] W. J. Ke and S. D. Wang, "Reliability evaluation for distributed computing networks with imperfect nodes", IEEE Transactions on Reliability, vol. 46, no. 3, pp. 342-349, 1997.
[9] C. J. Colbourn, "The combinatorics of network reliability", Oxford University Press, Inc., 1987.
[10] W. C. Yeh, "A simple minimal path method for estimating the weighted multi-commodity multistate unreliable networks reliability", Reliability Engineering & System Safety, vol. 93, no. 1, pp. 125-136, 2008.
[11] J. Nahman, "Minimal paths and cuts of networks exposed to common-cause failures", IEEE Transactions on Reliability, vol. 41, no. 1, pp. 76-80, 1992.
[12] Z. Tan, "Minimal cut sets of s–t networks with k-out-of-n nodes", Reliability Engineering & System Safety, vol. 82, no. 1, pp. 49-54, 2003.
[13] M. J. Zuo and M. Liang, "Reliability of multistate consecutively-connected systems", Reliability Engineering & System Safety, vol. 44, no. 2, pp. 173-176, 1994.
[14] W. C. Yeh, "Multistate-node acyclic networks reliability evaluation based on MC", Reliability Engineering & System Safety, vol. 81, no. 2, pp. 225-231, 2003.
[15] U. Buscher and G. Lindner, "Optimizing a production system with rework and equal sized batch shipments", Computers & Operations Research, vol. 34, no. 2, pp. 515-535, 2007.
[16] R. Teunter, K. Kaparis, and O. Tang, "Multi-product economic lot scheduling problem with separate production lines for manufacturing and remanufacturing", European journal of operational research, vol. 191, no. 3, pp. 1241-1253, 2008.
[17] P. C. Chang, "Reliability of a maintainable manufacturing network subject to budget", Int. J. Oper. Res., vol. 13, no. 3, pp. 95-101, 2016.
[18] Y. K. Lin and P. C. Chang, "A novel reliability evaluation technique for stochastic-flow manufacturing networks with multiple production lines", IEEE Transactions on Reliability, vol. 62, no. 1, pp. 92-104, 2013.
[19] Y. K. Lin and P. C. Chang, "Evaluate the system reliability for a manufacturing network with reworking actions", Reliability Engineering & System Safety, vol. 106, pp. 127-137, 2012.
[20] E. F. Moore and C. E. Shannon, "Reliable circuits using less reliable relays", Journal of the Franklin Institute, vol. 262, no. 3, pp. 191-208, 1956.
[21] X. Janan, "On multistate system analysis", IEEE Transactions on Reliability, vol. 34, no. 4, pp. 329-337, 1985.
[22] W. C. Yeh, "Multistate network reliability evaluation under the maintenance cost constraint", International Journal of Production Economics, vol. 88, no. 1, pp. 73-83, 2004.
[23] T. Aven and R. Østebø, "Two new component importance measures for a flow network system", Reliability Engineering, vol. 14, no. 1, pp. 75-80, 1986.
[24] Y. K. Lin and C. T. Yeh, "Computer network reliability optimization under double-resource assignments subject to a transmission budget", Information Sciences, vol. 181, no. 3, pp. 582-599, 2011.
[25] M. J. Zuo, Z. Tian, and H. Z. Huang, "An efficient method for reliability evaluation of multistate networks given all minimal path vectors", IIE transactions, vol. 39, no. 8, pp. 811-817, 2007.
[26] E. A. Bender and S. G. Williamson, "Lists, Decisions and Graphs", S. Gill Williamson, 2010.
[27] M. E. Newman, "The structure and function of complex networks", SIAM review, vol. 45, no. 2, pp. 167-256, 2003.
[28] H. Lee and A. Garcia-Diaz, "A network flow approach to solve clustering problems in group technology", International Journal of Production Research, vol. 31, no. 3, pp. 603-612, 1993.
[29] H. Lee and A. Garcia-Diaz, "Network flow procedures for the analysis of cellular manufacturing systems", IIE transactions, vol. 28, no. 4, pp. 333-345, 1996.
[30] A. Vereecke, R. Van Dierdonck, and A. De Meyer, "A typology of plants in global manufacturing networks", Management Science, vol. 52, no. 11, pp. 1737-1750, 2006.
[31] B. Sager, S. Hawer, and G. Reinhart, "A Performance Measurement System for Global Manufacturing Networks", Procedia CIRP, vol. 57, pp. 61-66, 2016.
[32] A. M. Zargar, "Effect of rework strategies on cycle time", Computers & Industrial Engineering, vol. 29, no. 1-4, pp. 239-243, 1995.
[33] Y. K. Lin and P. C. Chang, "System reliability of a manufacturing network with reworking action and different failure rates", International Journal of Production Research, vol. 50, no. 23, pp. 6930-6944, 2012.
[34] L. Fiondella, Y. K. Lin, and P. C. Chang, "System performance and reliability modeling of a stochastic-flow production network: a confidence-based approach", IEEE Transactions on Systems, Man, Cybernetics: Systems, vol. 45, no. 11, pp. 1437-1447, 2015.
[35] F. L. Zhou, X. Wang, and Y. Lin, "Production effectiveness-based system reliability calculation of serial manufacturing with checking machine", 電腦學刊, vol. 27, no. 3, pp. 201-211, 2016.
[36] D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, "Why the Monte Carlo method is so important today", Wiley Interdisciplinary Reviews: Computational Statistics, vol. 6, no. 6, pp. 386-392, 2014.
[37] W. K. Hastings, "Monte Carlo sampling methods using Markov chains and their applications", Biometrika, vol. 57, no. 1, pp. 97-109, 1970.
[38] N. Metropolis, "The beginning of the Monte Carlo method", Los Alamos Science, vol. 15, no. 584, pp. 125-130, 1987.
[39] M. Ljungberg, S. E. Strand, and M. A. King, "Monte Carlo calculations in nuclear medicine: Applications in diagnostic imaging", CRC Press, 2012.
[40] M. Mesta et al., "Molecular-scale simulation of electroluminescence in a multilayer white organic light-emitting diode", Nature materials, vol. 12, no. 7, p. 652, 2013.
[41] O. Stenzel, L. J. A. Koster, R. Thiedmann, S. D. Oosterhout, R. A. Janssen, and V. Schmidt, "A new approach to model‐based simulation of disordered polymer blend solar cells", Advanced Functional Materials, vol. 22, no. 6, pp. 1236-1244, 2012.
[42] G. Fishman, "Monte Carlo: concepts, algorithms, and applications", Springer Science & Business Media, 2013.
[43] D. P. Kroese, T. Taimre, and Z. I. Botev, "Handbook of monte carlo methods", John Wiley & Sons, 2013.
[44] H. M. Choset et al., "Principles of robot motion: theory, algorithms, and implementation", MIT press, 2005.
[45] T. Gerstner and P. E. Kloeden, "Recent developments in computational finance: foundations, algorithms and applications", World Scientific, 2013.
[46] R. H. Swendsen and J. S. Wang, "Nonuniversal critical dynamics in Monte Carlo simulations", Physical review letters, vol. 58, no. 2, p. 86, 1987.
[47] S. Wheelwright, S. Makridakis, and R. J. Hyndman, "Forecasting: methods and applications", John Wiley & Sons, 1998.
[48] C. D. Lewis, "Industrial and business forecasting methods: A practical guide to exponential smoothing and curve fitting", Butterworth-Heinemann, 1982.