本研究為一案例公司的生產線問題，該公司生產四個產品並有三個工作站：組裝，檢查和分解工作站。新產品和維修產品都在同一生產線上加工，然而，它們有不同的生產路線。最終的檢測工作站加工時間最長，且時間浮動較大。此問題即為一混合式流程生產（Flexible flow shop）的隨機排程問題，案例公司想提高準時交貨率，但由於工作的加工時間是不確定的，因此其完成時間也是不確定的；延遲的時間也無法確定。為了準時交貨，本研究旨在透過調換每個工作站對工作加工的優先順序，盡量減少期望的總加權延遲時間。
為了解決這個問題，本研究利用了離散事件模擬系統，其中考慮了隨機加工時間，每個工作站上對每個產品的加工時間服從一經驗分佈：此外，還考慮了重工和機器故障的可能。在本研究中，求得的解是指所有工作站的加工優先順序列表。給定一個解，經過充分次數的模擬，可以得到每個工作完成時間的分佈，如此一來，就可以計算每個工作的期望完工時間以及所有工作的期望總加權延遲時間，即為目標值。
本研究採用迭代平行模擬退火法尋找改善解，該方法利用多執行緒加速求解的效率，因此在同樣的時間內，可以找更多可行解，也就能越快找到最佳解。從一開始用最早交期派工法則排出起始解後，透過第一次平行模擬退火法找出最佳解，再將該解當成第二次平行模擬退火法的起始解，繼續尋找最佳解，最後輸出各個工作的預期完工時間；以及最佳解。最後將結果與最早交期派工法則比較，能發現大約都有10%或以上的改善率，且配對t檢定證明在99%的信心水準下，其差異為顯著的。

This study is based on the actual production of a case company, in which four products are produced. There are three workstations in the factory: assembly, inspection, and decomposition workstations. Both new products and repaired ones are processed in the same factory; however, their routes are different. The processing time at final inspection workstation is longest and has a large variation. This problem can be regarded as a flexible flow shop scheduling problem with stochastic processing times. The case company would like to increase its on-time delivery rate. However, due to the unreliability of the final inspection process, predicting the completion and thus meeting the due date are difficult tasks. The processing time of a job is uncertain and therefore its completion time is also uncertain. Hence, its tardiness is also uncertain. To improve on-time delivery, this study aims to calculate a job dispatching priority for each workstation with the objective of minimizing the expected total weighted tardiness.
To tackle this issue with the availability of the empirical distribution of the processing time of each product on each workstation, this study utilizes the discrete event simulation, in which random processing times, reworks and machine breakdowns are considered. In this study, a solution to a particular scheduling instance refers to the dispatching priority lists for all workstations. Given a solution, after a sufficient replicates of simulations, the empirical distribution of completion time for each job can be obtained. Then, the expected tardiness of each job and thus the expected total tardiness of all jobs, the objective function value, can be calculated. Heuristic search method explores various solution spaces and search for a better solution that minimizes the expected tardiness.
In this study, an iterative parallel simulated annealing method is used to find the optimal solution. This method uses multiple threads to speed up the solution, and therefore, in the same time, more feasible solutions can be found, which means the best solution can be found faster. Take earliest due date dispatching rule as initail solution for first time SA. After the whole complete round, the optimal solution was output to next SA as initial solution. The result will be best job priority list in each workstation and expected completion time of each job. Comparing with the job priority list generated by earliest due date rule, the improvement rate is about 10% or more, and the difference is significant at 99% confidence level under paired t-test, which suggest the method we introduced is better.

1. 緒論 1
1.1 研究背景 1
1.2 研究動機 2
1.3 問題描述 2
1.4 研究方法 3
2. 文獻回顧 5
2.1 流程式生產 5
2.1.1 排程問題環境 5
2.1.2 混合式流程生產 6
2.2 重工 7
2.3 隨機排程問題 8
2.3.1 隨機加工時間 8
2.3.2 搜尋法 8
2.4 平行機台排程問題 9
3. 方法建構 11
3.1 問題假設以及求解流程 11
3.1.1 問題假設 11
3.1.2 求解流程 12
3.2 模擬系統內部設置 14
3.2.1 隨機加工時間 14
3.2.2 重工 15
3.2.3 機器故障 16
3.3 平行模擬退火演算法 16
3.3.1 模擬退火法符號定義 17
3.3.2 模擬退火法流程 18
3.3.3 平行模擬退火法 21
3.3.4 迭代平行模擬退火法 23
3.4 模擬結果之資料處理 25
4. 實驗設計與分析 26
4.1 實驗設計 26
4.2 參數設定 27
4.3 產生問題 31
4.4 實驗結果與分析 32
4.4.1 平行模擬退火法與最早交期派工法則比較 32
4.4.2 各因子在不同水準下比較 34
5. 結論與未來展望 39
參考文獻 40

Arthanari, T.S. and Ramamurthy, K.G., (1971), “Extension of two machines sequencing problem,” Opsearch 8, 10–22.
AitZai, A., and Boudhar, M., (2013), “Parallel branch-and-bound and parallel PSO algorithms for job shop scheduling problem with blocking,” International Journal of Operational Research, Vol. 16, Issue. 1, pp. 14-37.
AitZai, A., Boudhar, M., and Dabah, A., (2013), “Parallel CPU and GPU Computations to Solve the Job Shop Scheduling Problem with Blocking,” IEEE High Performance Extreme Computing Conference, Waltham, USA.
Akhshabi, M., Haddadnia, J., and Akhshabi, M., (2012), “Solving flow shop scheduling problem using a parallel genetic algorithm,” Procedia Technology, Vol. 1, pp. 351-355.
Anghinolfi, D., and Paolucci, M., (2007), “Parallel machine total tardiness scheduling with a new hybrid. metaheuristic approach,” Computers & Operations Research, Vol. 34, No. 11, pp. 3471–3490.
Bootaki, B. and Paydar, M.M. (2017), “On the n-job, m-machine permutation flow shop scheduling problems with makespan criterion and rework,” Scientia Iranica, doi: 10.24200/sci.2017.4443.
Bożejko, W., and Wodecki, M., (2002), “Solving the flow shop problem by parallel tabu search,” Parallel Computing in Electrical Engineering, Proceedings. International Conference, IEEE, pp. 189-194.
Bożejko, W., and Wodecki, M., (2003), “Parallel genetic algorithm for the flow shop scheduling problem,” International Conference on Parallel Processing and Applied Mathematics, Heidelberg, GERMANY, pp. 566-571.
Bożejko, W., and Wodecki, M., (2008). “Parallel scatter search algorithm for the flow shop sequencing problem,” Parallel Processing and Applied Mathematics, pp. 180-188.
Bożejko, W., and Wodecki, M., (2011), “The methodology of parallel memetic algorithms designing,” ICAART-2011 3rd International Conference on Agents and Artificial Intelligence, Rome, Italy, pp. 643-648.
Chandrasekharan R., and Oliver H., (1999), “A comparative study of dispatching rules in dynamic flowshops and jobshops,” European Journal of Operational Research, Vol. 116, Issue 1, pp. 156-170.
Defersha, F. M., and Chen, M., (2010), “A parallel genetic algorithm for a flexible job-shop scheduling problem with sequence dependent setups,” The international journal of advanced manufacturing technology, Vol. 49, Issue 1, pp. 263-279.
Eskandari, H. and Hosseinzadeh, A., (2014). “A variable neighbourhood search for hybrid flow-shop scheduling problem with rework and set-up times,” Journal of the Operational Research Society. Vol. 65, Issue 8, pp. 1221-1231.
Flapper, S.D.P., Fransoo, J.C., Broekmeulen, R.A.C.M. and Inderfurth, K., (2002), “Planning and control of rework in the process industries: a review,” Production Planning & Control, Vol. 1, pp. 26–34.
Gao, J., (2005), ” A Parallel Hybrid Genetic Algorithm for Solving a Kind of Non-Identical Parallel Machine Scheduling Problems,” 2005 IEEE High-Performance Computing Conference, Beijing, China, pp. 469-472.
Gao, J., He, G., and Wang, Y., (2009), “A new parallel genetic algorithm for solving multiobjective scheduling problems subjected to special process constraint,” The International Journal of Advanced Manufacturing Technology, Vol. 43, Issue. 1, pp. 151-160.
Graham, R.L. (1979), “Optimization and Approximation in Deterministic Sequencing and Scheduling: a survey,” Ann. Discrete Math, 5, Pages 287-326.
Inderfurth, K. and Teunter, R.H. (2003), “Production planning and control of closed-loop supply chains,” Carnegie Mellon University Press, Oxford, pp. 149–173.
Hung, Y. F., Chen, W. C., and Chen, J. C., (2012), “Search algorithms in the selection of warehouses and transshipment arrangement for high-end low-volume products,” Journal of Advanced Engineering, Vol. 7, No. 2, pp. 51-60.
Xu, J.Y, Wu, C.C., Yin, Y.Q., and Lin, W.C. (2017), “An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times,” Applied Soft Computing, Vol. 52, pp. 39-47.
Hunsucker, J.L., and Shah, J.R. (1994), “Comparative performance analysis of priority rules in a constrained flow shop with multiple processors environment,” European Journal of Operational Research, Vol. 72, Issue 1, pp. 102-114.
Wang, K., Choi, S.H., Qin, H., and Huang, Y. (2013), “A cluster-based scheduling model using SPT and SA for dynamic hybrid flow shop problems,” The International Journal of Advanced Manufacturing Technology, Vol. 67, pp. 2243.
Lee, C. Y., and Kim, S. J., (1995), ”Parallel genetic algorithms for the earliness-tardiness job scheduling problem with general penalty weights,” Computers & Industrial Engineering, Vol. 28, No. 2, pp. 231-243.
Lee, W.C., Wu, C.C., and Chen, P., (2006), “A simulated annealing approach to makespan minimization on identical parallel machines,” International Journal of Advanced Manufacturing Technology, Vol. 31, pp. 328-334.
Lundy, M., and Mess, A., (1986), “Convergence of an annealing algorithm,” Mathematical Programming, Vol. 34, pp. 111-124.
Kirkpatrick, S., Gelatt, J.R., and Vecchi, M.P., (1983), “Optimization by simulated annealing,” Science, Vol. 200, No. 4598, pp. 671–680.
Kim, D.W., Kim, K. H., Jang, W., and Chen, F.F., (2002), “Unrelated parallel machine scheduling with setup times using simulated annealing,” Robotics and Computer Integrated Manufacturing, Vol. 18, pp. 223–231.
Rabiee, M., Zandieh M., and Jafarian, A., (2012), “Scheduling of a no-wait two-machine flow shop with sequence-dependent setup times and probable rework using robust meta-heuristics,” International Journal of Production Research, Vol. 50, Issue 24, Pages 7428-7446.
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. and Teller, E., (1953), “Equation of state calculations by fast computing machines,” Journal of Chemical Physics, Vol. 21, No. 6, pp. 1087–1092.
Matsveichuk, N.M., Sotskov, Y.N., Egorova, N.G., and Lai, T.C., (2009), “Schedule execution for two-machine flow-shop with interval processing times,” Mathematical and Computer Modelling, Vol. 49, Issues 5-6, pp. 991-1011.
Park, M.W., and Kim, Y.D., (1997), “Search heuristics for a parallel machine scheduling problem with ready times and due dates,” Computers & Operations Research, Vol. 33, Issue. 3-4, pp. 793-796.
Ku, P.S., and Niu, S.C., (1986), “On Johnson's two-machine flow shop with random processing times,” Operation Research, Vol. 34, Issue 1, pp. 130-136.
Perregaard, M., and Clausen, J., (1998), “Parallel branch-and-bound methods for the job-shop scheduling problem,” Annals of Operations Research, Vol. 83, pp. 137-160.
Pinedo, M. (1995), “Scheduling Theory, Algorithm, and System,” Prentice Hall, NEW Jersey.
Rashidi, E., Jahandar, M., and Zandieh, M., (2010), “An improved hybrid multi-objective parallel genetic algorithm for hybrid flow shop scheduling with unrelated parallel machines”, The International Journal of Advanced Manufacturing Technology, Vol. 49, Issue. 9, pp. 1129-1139.
Pan, Q.K., and Ruiz, R., (2012), “Local search methods for the flowshop scheduling problem with flowtime minimization,” European Journal of Operational Research, Vol. 222, Issue 1,pp. 31-43.
Rabiee, M., Jolai, F., and Gheisariha, E. (2014). “An enhanced invasive weed optimization for makespan minimization in a flexible flowshop scheduling problem,” Scientia Iranica. Vol. 21, Issue 3, pp. 1007-1020.
Elmaghraby, S.E., and Thoney K.A., (1999), “The two-machine stochastic flowshop problem with arbitrary processing time distributions,” IIE Transactions, Vol. 31, Issue 5, pp. 467-477.
Shahram E., and Saeedeh G., (2015), “Multi-objective flexible flow shop scheduling with unexpected arrivals of new jobs,” Applied mathematics in Engineering, Management and Technology, Vol. 3, Issue 3, pp. 172-181
Choi, S.H., and Wang, K., (2012), “Flexible flow shop scheduling with stochastic processing times: A decomposition-based approach,” Computers & Industrial Engineering, Vol. 63, Issue 2, pp. 362-373.
Thiruvady, D., Ernst, A. T., and Singh, G., (2016), “Parallel ant colony optimization for resource constrained job scheduling,” Annals of Operations Research, Vol. 2, No. 242, pp. 355-372.
Huang, W.Q., and Wang, L. (2006), “A Local Search Method for Permutation Flow Shop Scheduling,” The Journal of the Operational Research Society, Vol. 57, Issue 10, pp. 1248-1251.