本研究以粒子群演算法(Particle Swarm Optimization, PSO)為基礎，結合Ranking and Selection (R&S)方法求解具單一隨機限制式之離散型模擬最佳化問題。在解決此類問題時，需要在有限的時間或預算之下，找出最佳或近似最佳方案。由於問題之限制存在隨機性，故在解空間中同時存在許多可行解與不可行解。在處理此類問題時，過去學者使用啟發式演算法解決方案數過多的問題；以R&S方法解決抽樣資源分配問題。但仍沒有方法可同時考量方案數過多、方案是否可行以及抽樣資源分配等因子，而導致該問題在求解的過程中產生模擬耗時且效率不佳的情況。本研究以Optimal Simulation Budget Allocation for Constrained Optimization (OCBA-CO)為基礎，提出一最佳抽樣分配策略(Optimal Sample Allocation Strategy for Constrained Optimization, OSAS-CO)。考量方案的變異性以及可行性，並加入Super individual以及菁英解族群(Elite group)的概念，將資源分配在關鍵方案上，提升選到最佳方案的機率，再將其應用於PSO，形成一混合式演算法。
透過PSO之特性，改善方案數過多所產生之模擬耗時的問題；透過OSAS-CO之特性，在分配重覆模擬次數的同時評估方案之可行性，並改善PSO結合OCBA-CO時可能會耗費模擬計算資源在績效相近的方案上之問題，提升抽樣資源分配的效率。以此混合式演算法求解具單一隨機限制之模擬最佳化問題。
本研究分別使用兩種不同特性之函數模型以及緩衝區分配問題，並於這三種實驗中分別減少57%、14.4%以及21.96%使用的總模擬次數，證實OSAS-CO能夠顯著改善模擬計算資源使用之效率，有效減少使用的總模擬次數。

This research is based on Particle Swarm Optimization (PSO) and combine with Ranking and Selection (R&S) method to solve the discrete event simulation optimization problem with single stochastic constraint. It needs to find the optimal or nearly optimal solution in finite time or budget when solving these problems. Because the constraints of problem exist stochastic, there are many feasible and infeasible solutions existing in solution space simultaneously. When dealing with such issues, researchers used heuristic algorithms to solve the problem of excessive number of solutions, and used R&S method to solve the sampling resource allocation problem in the past. However, there is still no method to consider the factors such as excessive number of plans, feasibility of plans, and sampling resource allocation simultaneously. This makes the problem with time-consuming in simulation and inefficient in solution solving process. Based on Optimal Simulation Budget Allocation for Constrained Optimization (OCBA-CO), this research proposed an Optimal Sample Allocation Strategy for Constrained Optimization (OSAS-CO) method. This method considers the variability and feasibility of solutions, and adds the concepts of Super individual and Elite group to allocate resources on key solutions for increasing the probability of selecting the best solution. Then applied to PSO to construct a hybrid algorithm.
According to the characteristics of PSO, the proposed method can improve the problem of excessive number of solutions that makes simulation time consuming. According to the characteristics of OSAS-CO, our method can also evaluate the feasibility of solutions while allocate the repetitive simulations. Then it improves the problem that PSO combines OCBA-CO will consume a lot of simulation computation budgets on solutions which have similar performance, and increase the sampling resource allocation efficiency. This research used the hybrid algorithm to solve the simulation optimization problem with single stochastic constraint.
This research used two different functional models and the buffer allocation problems respectively, and reduced the total simulation number by 57%, 14.4%, and 21.96% in three experiments respectively. This research proved that OSAS-CO can improve the usage efficiency of simulation computation budgets and reduce the total simulation numbers significantly.

1. Ahmed, M. A., & Alkhamis, T. M. (2009). Simulation optimization for an emergency department healthcare unit in Kuwait. European Journal of Operational Research, 198(3), 936-942.
2. Andradóttir, S., Goldsman, D., & Kim, S. H. (2005, December). Finding the best in the presence of a stochastic constraint. In Proceedings of the 37th Conference on Winter Simulation (pp. 732-738). Winter Simulation Conference.
3. Andradóttir, S., & Kim, S. H. (2010). Fully sequential procedures for comparing constrained systems via simulation. Naval Research Logistics, 57(5), 403-421.
4. Baesler, F. F., Jahnsen, H. E., & DaCosta, M. (2003, December). Emergency departments I: the use of simulation and design of experiments for estimating maximum capacity in an emergency room. In Proceedings of the 35th Conference on Winter Simulation: Driving innovation (pp. 1903-1906). Winter Simulation Conference.
5. Banks, A., Vincent, J., & Anyakoha, C. (2008). A review of particle swarm optimization. Part II: Hybridisation, combinatorial, multicriteria and constrained optimization, and indicative applications. Natural Computing, 7(1), 109-124.
6. Barton, R. R., & Meckesheimer, M. (2006). Metamodel-based simulation optimization. Handbooks in Operations Research and Management Science, 13, 535-574.
7. Batur, D., & Kim, S. H. (2005, December). Procedures for feasibility detection in the presence of multiple constraints. In Proceedings of the 37th Conference on Winter Simulation (pp. 692-698). Winter Simulation Conference.
8. Chaharsoughi, S., & Nahavandi, N. (2003). Buffer allocation problem, a heuristic approach.
9. Bechhofer, R. E., Santner, T. J., & Goldsman, D. M. (1995). Design and analysis of experiments for statistical selection, screening, and multiple comparisons. Wiley, New York.
10. Chen, C. H., Lin, J., Yücesan, E., & Chick, S. E. (2000). Simulation budget allocation for further enhancing the efficiency of ordinal optimization. Discrete Event Dynamic Systems, 10(3), 251-270.
11. Chen, C. H., & Lee, L. H. (2011). Stochastic simulation optimization: An optimal computing budget allocation (Vol. 1). World scientific.
12. Chen, W., Gao, S., Chen, C. H., & Shi, L. (2014). An optimal sample allocation strategy for partition-based random search. IEEE Transactions on Automation Science and Engineering, 11(1), 177-186.
13. Chiu, C. C., & Lin, J. T. (2017). Novel hybrid approach with elite group optimal computing budget allocation for the stochastic multimodal problem. Neurocomputing, 260, 449-465.
14. Clerc, M., & Kennedy, J. (2002). The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE Transactions on Evolutionary Computation, 6(1), 58-73.
15. Coello, C. A. C. (2000). Use of a self-adaptive penalty approach for engineering optimization problems. Computers in Industry, 41(2), 113-127.
16. Coello, C. A. C., & Montes, E. M. (2002). Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Advanced Engineering Informatics, 16(3), 193-203.
17. Cruz, F. R., Duarte, A. R., & Van Woensel, T. (2008). Buffer allocation in general single-server queueing networks. Computers & Operations Research, 35(11), 3581-3598.
18. Demir, L., Tunali, S., & Eliiyi, D. T. (2014). The state of the art on buffer allocation problem: a comprehensive survey. Journal of Intelligent Manufacturing, 25(3), 371-392.
19. Dudewicz, E. J., & Dalal, S. R. (1975). Allocation of observations in ranking and selection with unequal variances. Sankhyā: The Indian Journal of Statistics, Series B, 28-78.
20. Eberhart, R., & Kennedy, J. (1995, October). A new optimizer using particle swarm theory. In Micro Machine and Human Science, 1995. MHS'95., Proceedings of the Sixth International Symposium on (pp. 39-43). IEEE.
21. Gao, S., & Chen, W. (2017). A partition-based random search for stochastic constrained optimization via simulation. IEEE Transactions on Automatic Control, 62(2), 740-752.
22. Gershwin, S. B., & Schor, J. E. (2000). Efficient algorithms for buffer space allocation. Annals of Operations Research, 93(1), 117-144.
23. Harris, J. H., & Powell, S. G. (1999). An algorithm for optimal buffer placement in reliable serial lines. IIE Transactions, 31(4), 287-302.
24. He, Q., & Wang, L. (2007). A hybrid particle swarm optimization with a feasibility-based rule for constrained optimization. Applied Mathematics and Computation, 186(2), 1407-1422.
25. Hunter, S. R., & Pasupathy, R. (2013). Optimal sampling laws for stochastically constrained simulation optimization on finite sets. INFORMS Journal on Computing, 25(3), 527-542.
26. Jeong, K. C., & Kim, Y. D. (2000). Heuristics for selecting machines and determining buffer capacities in assembly systems. Computers & industrial Engineering, 38(3), 341-360.
27. Kabirian, A., & Olafsson, S. (2009, December). Selection of the best with stochastic constraints. In Simulation Conference (WSC), Proceedings of the 2009 Winter (pp. 574-583). IEEE.
28. Kim, S., & Lee, H. J. (2001). Allocation of buffer capacity to minimize average work-in-process. Production Planning & Control, 12(7), 706-716.
29. Kim, S. H., & Nelson, B. L. (2007, December). Recent advances in ranking and selection. In Simulation Conference, 2007 Winter (pp. 162-172). IEEE.
30. Kleijnen, J. P., Van Beers, W., & Van Nieuwenhuyse, I. (2010). Constrained optimization in expensive simulation: Novel approach. European Journal of Operational Research, 202(1), 164-174.
31. Lee, L. H., Pujowidianto, N. A., Li, L. W., Chen, C., & Yap, C. M. (2012). Approximate simulation budget allocation for selecting the best design in the presence of stochastic constraints. IEEE Transactions on Automatic Control, 57(11), 2940-2945.
32. Luo, Y., & Lim, E. (2013). Simulation-based optimization over discrete sets with noisy constraints. IIE Transactions, 45(7), 699-715.
33. Matejcik, F. J., & Nelson, B. L. (1993, December). Simultaneous ranking, selection and multiple comparisons for simulation. In Proceedings of the 25th Conference on Winter Simulation (pp. 386-392). ACM.
34. Michalewicz, Z. (1995). A survey of constraint handling techniques in evolutionary computation methods. Evolutionary Programming, 4, 135-155.
35. Pan, H., Wang, L., & Liu, B. (2006). Particle swarm optimization for function optimization in noisy environment. Applied Mathematics and Computation, 181(2), 908-919.
36. Park, C., & Kim, S. H. (2015). Penalty function with memory for discrete optimization via simulation with stochastic constraints. Operations Research, 63(5), 1195-1212.
37. Pasupathy, R., & Nagaraj, K. (2015, December). Modeling dependence in simulation input: The case for copulas. In Winter Simulation Conference (WSC), 2015 (pp. 1850-1864). IEEE.
38. Pujowidianto, N. A., Lee, L. H., Chen, C. H., & Yap, C. M. (2009, December). Optimal computing budget allocation for constrained optimization. In Simulation Conference (WSC), Proceedings of the 2009 Winter (pp. 584-589). IEEE.
39. Pujowidianto, N. A., Pasupathy, R., Hunter, S. R., Lee, L. H., & Chen, C. H. (2012, December). Closed-form sampling laws for stochastically constrained simulation optimization on large finite sets. In Simulation Conference (WSC), Proceedings of the 2012 Winter (pp. 1-10). IEEE.
40. Rinott, Y. (1978). On two-stage selection procedures and related probability-inequalities. Communications in Statistics-Theory and Methods, 7(8), 799-811.
41. Runarsson, T. P., & Yao, X. (2000). Stochastic ranking for constrained evolutionary optimization. IEEE Transactions on Evolutionary Computation, 4(3), 284-294.
42. Shi, L., & Olafsson, S. (2009). Nested partitions method, theory and applications. New York: Springer.
43. Szechtman, R., & Yucesan, E. (2008, December). A new perspective on feasibility determination. In Simulation Conference, 2008. WSC 2008. Winter (pp. 273-280). IEEE.
44. Tsai, S. C., & Fu, S. Y. (2014). Genetic-algorithm-based simulation optimization considering a single stochastic constraint. European Journal of Operational Research, 236(1), 113-125.
45. Vieira, H., Kienitz, J. K. H., & Belderrain, M. C. N. (2011, December). Discrete-valued, stochastic-constrained simulation optimization with COMPASS. In Proceedings of the Winter Simulation Conference (pp. 4196-4205). Winter Simulation Conference.
46. Yamada, T., & Matsui, M. (2003). A management design approach to assembly line systems. International Journal of Production Economics, 84(2), 193-204.
47. Yamashita, H., & Altiok, T. (1998). Buffer capacity allocation for a desired throughput in production lines. IIE Transactions, 30(10), 883-891.
48. Yan, S., Zhou, E., & Chen, C. H. (2012). Efficient selection of a set of good enough designs with complexity preference. IEEE Transactions on Automation Science and Engineering, 9(3), 596-606.
49. Zhang, S., Chen, P., Lee, L. H., Peng, C. E., & Chen, C. H. (2011, December). Simulation optimization using the particle swarm optimization with optimal computing budget allocation. In Simulation Conference (WSC), Proceedings of the 2011 Winter (pp. 4298-4309). IEEE.
50. Zhang, J., Li, Z., Wang, C., Zang, D., & Zhou, M. (2015, May). Approximately Optimal Computing-Budget Allocation for Subset Ranking. In Robotics and Automation (ICRA), 2015 IEEE International Conference on (pp. 3856-3861). IEEE.