本研究考量兩階層存貨問題，存貨系統中包含一個中心倉庫以及多個不同零售商，整體供應鏈採用定期盤點策略，中心倉庫負責下游所有零售商的補貨訂單，且倉庫定期盤點存貨並向外部供應商下訂單，需求發生於零售商端為隨機變數，過往多數研究存貨問題的文獻為了分析上的可行或方便皆採用遇缺補貨假設居多，亦假設訂單前置期為定值，本研究認為銷售損失假設更能反映真實零售市場的環境，同時放寬前置期的限制，讓訂單前置期具有隨機性，主要目的為求出最大庫存水準以及盤點間隔時間，並使得各種存貨相關成本最小化。

由於本研究需求發生與前置期皆為隨機變數，造成此問題具有隨機目標式，且同時要求解包含離散型決策變數以及連續型決策變數，另外又擁有銷售損失假設，較難以利用解析解進行分析，因此提出結合隨機近似法 (Stochastic Approximation) 以及排序與選擇程序 (Ranking and Selection) 的混整數模擬最佳化演算法，即為 SASP 演算法，所有演算法迭代包含兩個階段，每次迭代皆產生數個候選整數解組合，給定起始連續型變數，先固定其離散型變數組合，並針對連續型變數執行隨機近似法階段，使不同的整數解組合下，搭配各自的連續型變數，接著進行目標式篩選程序階段，挑選出數個期望績效值較佳的候選解，不斷重複此兩個階段，在最終迭代若只有一個則為最佳解，如果有多個候選解則進行兩階段選擇程序，篩選出期望績效值最優的為最終最佳解。

本研究將 SASP 、 greedy 及 OptQuest 三個演算法進行比較所求得之最終解品質、使用的樣本數量以及整體演算法過程搜尋解的數量，實驗結果顯示在使用相近的樣本數下，SASP 找到的最終解比其他兩個演算法更優，此外，在未來研究方向提出一個具有發展潛力的方法。

We consider a time-base inventory control policy for a two-echelon inventory system consisting of a central warehouse and multiple non-identical retailers. In contrast to previous research, the two-echelon inventory system considered in our study with a lost-sales assumption and stochastic lead time is more realistic in many retail environments than its counterparts. The goal is to determine the order-up-to-level at the warehouse, retailers' order-up-to-level, and the inventory review interval for retailers that minimizes the expected total cost.

We develop a mixed-integer simulation optimization algorithm (SASP), which combines stochastic approximation with ranking and selection, to solve the problem. First, we generate multiple integer solutions and given their initial continuous solutions for every iteration. Second, for all integer solutions, we implement stochastic approximation on their corresponding continuous solution. Third, for all candidate solutions, we apply objective screening procedures with respect to the objective performance. Finally, we apply two-stage selection procedure to choose the best solution depend on the number of surviving solutions in last iteration.

This study compare the performance of final solution, the number of samples and the number of visited solutions among three different algorithms, SASP, greedy and OptQuest. The empirical results show that the final solution of SASP performs better than the others while three algorithms use similar number of samples.

蔡晉維 (2018) 。 以混合整數模擬最佳化求解具服務水準限制式之兩階層存貨問題。 國立成功大學工業與資訊管理系碩士論文。
Andersson, J., & Melchiors, P. (2001). A two-echelon inventory model with lost sales. International Journal of Production Economics, 69(3), 307-315.
Andradóttir, S. (2006). An overview of simulation optimization via random search. In: Henderson, S.G., & Nelson, B.L. (Eds.), Chapter 20 of Handbook in Operations Research and Management Science: Vol. 13. Simulation (pp. 617-631). North Holland, NL: Elsevier.
Bashyam, S., & Fu, M.C. (1998). Optimization of (s, S) inventory systems with random lead times and a service level constraint. Management Science, 44(12), 243- 256.
Bayliss, C., De Maere, G., Atkin, J.A.D., & Paelinck, M. (2017). A simulation scenario based mixed integer programming approach to airline reserve crew scheduling under uncertainty. Annals of Operations Research, 252(2), 335-363.
Bijvank, M., & Johansen, S.G. (2012). Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length. European Journal of Operational Research, 220, 106-114.
Bijvank, M., & Vis, I.F. (2011). Lost-sales inventory theory: A review. European Journal of Operational Research , 215, 1-13.
Boesel,J.,Nelson,B.L.,&Kim,S.H.(2003).Using ranking and selection to“clean up” after simulation optimization. Operations Research, 51(5), 814-825.
Chau, M., & Fu, M.C. (2015). An overview of stochastic approximation. In: Fu, M.C. (Eds.), Chapter 6 of Handbook of Simulation Optimization (pp. 149-178). New York, NY: Springer.
Chen, F., & Zheng, Y.S. (1997). One-warehouse multiretailer systems with centralized stock information. Operations Research, 45(2), 275-287.
Chick, S.E., & Inoue, K. (2001). New two-stage and sequential procedures for selecting the best simulated system. Operations Research, 49, 732-743.
Cheung, K.L., & Lee, H.L. (2002). The inventory benefit of shipment coordination and stock rebalancing in a supply chain. Management Science, 48(2), 300-306.
Feng, K., & Rao, U.S. (2007). Echelon-stock (R, nT) control in two-stage serial stochastic inventory systems. Operations Research Letters, 35, 95-104.
Fu, M.C., Glover, F.W., & April, J. (2005, December). Simulation optimization: A review, new developments, and applications. Paper presented at 2005 Winter Simulation Conference, Orlando, FL.
Gruen, T.W., Corsten, D.S., & Bharadwaj, S. (2002). Retail out-of-stocks: A world- wide examination of extent causes and consumer responses. Washington, DC: Grocery Manufacturers of America.
Hill, R.M., Seifbarghy, M., & Smith, D.K. (2007). A two-echelon inventory model with lost sales. European Journal of Operational Research. 181(2), 753-766.
Huh, W.T., Janakiraman, G., Muckstadt, J.A., & Rusmevichientong, P. (2009). Asymptotic optimality of order-up-to policies in lost sales inventory systems. Management Science, 55(3), 404-420.
Kim, S. H., & Nelson, B. L. (2001). A fully sequential procedure for indifference- zone selection in simulation. ACM Transactions on Modeling and Computer Simulation (TOMACS), 11(3), 251-273.
Li, J.C., Gong, B., & Wang, H. (2016). Mixed integer simulation optimization for optimal hydraulic fracturing and production of shale gas fields. Engineering Optimization, 48(8), 1378–1400.
Nelson, B.L., Swann, J., Goldsman, D., & Song, W. (2001). Simple procedures for selecting the best simulated system when the number of alternatives is large. Operations Research, 49(6), 950–963.
Özdemir, D., Yücesan, E., & Herer, Y.T., (2013). Multi-location transshipment problem with capacitated production. European Journal of Operational Research, 226(3), 425-435.
Schwarz, L.B., (1989). A model for assessing the value of warehouse risk-pooling: risk-pooling over outside-supplier lead times. Management Science, 35(7), 828-842.
Sezen, B., (2006). Changes in performance under various lengths of review periods in a periodic review inventory control system with lost sales: A simulation study. International Journal of Physical Distribution Logistics Management, 36(5), 360- 373.
Sherbrooke, C.C. (1968). METRIC: A multi-echelon technique for recoverable item control. Operations Research, 16(1), 122-141.
Sriver, T. A., Chrissis, J. W., & Abramson, M. A. (2009) Pattern search ranking and selection algorithms for mixed variable simulation-based optimization. European Journal of Operational Research, 198, 878-890.
Truong, T. H., & Azadivar, F. (2003, December). Simulation based optimization for supply chain configuration design. Proceedings of the Winter Simulation Conference 2003, New Orleans, Louisiana.
Wang, H. (2013, December). Mixed integer simulation optimization for petroleum field development under geological uncertainty. Proceedings of the Winter Simulation Conference 2013, Washington, DC.
Wang, Q. (2013). A periodic-review inventory control policy for a two-level supply chain with multiple retailers and stochastic demand. European Journal of Operational Research, 230(1), 53-62.
Wang, Q., & Axsäter, S. (2010). Fixed-interval joint-replenishment policies for distribution systems with multiple retailers and stochastic demand. Naval Research Logistics, 60(8), 637-651.