多階層存貨系統(Multi-echelon Inventory System)問題不論在應用上或是理論上一直是生產管理領域中一個重要的議題。從Sherbrooke(1968)提出多階層可維修備用件庫存模式(Multi-Echelon Technique for Recoverable Item Control; METRIC)以來，眾多學者致力於此議題的研究。從其豐富的文獻中，可窺見此議題重要的程度。本篇研究考慮一個二階層可維修商品庫存系統問題，在這個存貨系統中有一個總倉庫和多個服務站。當顧客的機器零件發生損壞時，服務站必須為顧客更換好的零件，並依據(S-1,S)存貨政策由總倉庫進行補貨。總倉庫同時也是維修中心，負責維修從服務站送回的損壞零件。我們希望找出總倉庫及服務站基本存量的最佳配置，在滿足各個服務站反應時間的門檻值下使整個系統的存貨投資成本最低。

排序與選擇程序(Ranking and Selection; R&S)主要用來處理解空間較小的問題，因為本研究所探討的問題由一個確切形式的目標式和多條隨機限制式所構成且具有龐大的解空間，若使用排序與選擇程序來處理則可能花費龐大的樣本數且解的品質不佳。另一方面，在既有的相關文獻中，多是以等候理論為基礎所發展的近似方法來處理多階層存貨系統問題，其缺點在於近似的過程中可能使得解產生誤差或為不可行解。因此，本研究發展了一個結合樣本平均近似法(Sample Average Approximation; SAA)及排序與選擇程序的模擬最佳化演算法來處理此問題。我們的演算法在每一次迭代都包含兩個階段，階段一利用切面法求解樣本問題，階段二則利用排序與選擇程序來判定階段一所得到的解是否為可行解。在最後實驗分析中將我們的演算法和樣本平均近似法及排序與選擇程序做比較，以佐證本研究確實能保證其解的可行性。

We study a two-echelon repairable inventory system, which consists of a central warehouse and multiple field depots that stock spare parts. When a failure occurs, the field depots serve the customers with part replacement and replenish their inventory from the central warehouse, following a base stock policy. The central warehouse also acts as a repair facility, and replenishes its inventory by repairing the defective parts passed by the field depots. Our goal is to find the best stocking levels in the central warehouse and field depots to minimize the system-wide inventory investment while maintaining an acceptable level of the expected response time over multiple field depots.

In our study, the time to failure, transportation time and repair time are random variables, and thus we formulate a problem with deterministic objective function but stochastic constraints. The problem we formulate has relatively large solution space; however, ranking and selection procedure (R&S) is mainly applied for the problem with a small number of solutions. Hence, we propose a simulation optimization algorithm that combines sample average approximation (SAA) and R&S to solve the problem. Our approach has two phases in each iteration. In Phase I, we obtain an optimal solution to the sample average version of the problem by applying linear programming and cutting-plane method. In Phase II, we employ R&S to check the feasibility of the solution. The samples we obtained in Phase II are stored and reused to perform Phase I in next iteration.

We integrate SAA and R&S in our approach, which has never been proposed in solving simulation optimization problems to our knowledge. We provide numerical results to show the efficiency of our proposed method.

Andrad´ottir, S., 1998. Simulation Optimization. In: Banks, J. (ED.), Handbook of Simulation, Chapter 9. John Wiley & Sons, New York, pp. 307-333.

Andrad´ottir, S., Kim, S.-H., 2010. Fully sequential procedures for comparing constrained systems via simulation. Naval Research Logistics 57, 403–421.

Atlason, J., Epelman, M.A., Henderson, S.G., 2004. Call center staffing with simulation and cutting plane methods. Annals of Operations Research 127, 333–358.

Atlason, J., Epelman, M.A., Henderson, S.G., 2008. Optimizing call center staffing using simulation and analytic center cutting-plane methods. Management Science 54 (2), 295–309.

Axs¨ater, S., 1993. Continuous review policies for multi-level inventory systems with stochastic demand. In: Graves, S.C., RinnooyKan, A.H.G. , Zipkin, P.H. (ED.), Logistics of Production and Inventory. North-Holland, New York, pp. 175–197.

Batur, D., Kim S.-H., 2010. Finding feasible systems in the presence of constraints on multiple performance measures. ACM Transactions on Modeling and Computer Simulation 20 (3), 1–26.

Bayraksan, G., Morton, D.P., 2010. A sequential sampling procedure for stochastic programming. Operations Research. Forthcoming.

Bazaraa, M.S., Sherali, H.D., Shetty, C.M., 2006. Nonlinear Programming: Theory and Algorithms, 3rd Edition. John Wiley & Sons, New York.

Bechhofer, R.E., 1954. A Single-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with Known Variances. Annals of Mathematical Statistics 25 , pp. 16-39.

Caggiano, K.E., Jackson, P.L., Muckstadt, J.A., Rappold, J.A., 2007. Optimizing service parts inventory in a multi-echelon, multi-item supply chain with time-based customer service level agreements. Operations Research 55 (2), 303–318.

Caggiano, K.E., Jackson, P.L., Muckstadt, J.A., Rappold, J.A. , 2009. Efficient computation of time-based customer service levels in a multi-item, multi-echelon supply chain: A practical approach for inventory optimization. European Journal of Operational Research 199, 744-749.

Caglar, D., 2001. A multi-echelon spare parts inventory system with emergency lateral shipments subject to a response time constraint. PhD dissertation, Northwestern University, Evanston, IL.

Caglar, D., Li, C.-L., Simchi-Levi, D., 2004. Two-echelon spare parts inventory system subject to a service constraint. IIE Transactions 36, 655–666.

Candasa, M.F., Kutanoglua E., 2007. Benefits of considering inventory in service parts logistics network design problems with time-based service constraints. IIE Transactions 39, 159–176.

Cattani, K.D., Jacobs, F. R., Schoenfelder, J., 2011. Common inventory modeling assumptions that fallshort:Arborescent networks, Poisson demand, and single-echelon approximations. Journal of Operations Management, 1–12.

Cezik, M.T., L’Ecuyer, P., 2008. Staffing multiskill call centers via linear programming and simulation. Management Science 54 (2), 310–323.

Cohen, M.A., Cull, C., Lee, H.L., Willen, D., 2000. Saturn’s supply chain innovation: high value in after sales service. MIT Sloan Management Review 41 (4), 93–101.

Cohen, M.A., Kleindorfer, P.R., Lee, H.L., 1988. Service constrained (s,S) inventory systems with priority demand classes and lost sales. Management Science 34 (4), 482–499.

Cohen, M.A., Zheng, Y.-S., Wang, Y., 1999. Identifying opportunities for improving Teradyne’s service parts logistics system. Interfaces 29 (4), 1–18.

Dai, L., Chen, C.H., Birge, J.R., 2000. Convergence Properties of Two-Stage Stochastic Programming. Journal of Optimization Theory and Applications 106 (3), 489–509.

Graves, S.C., 1985. A multi-echelon inventory model for a repairable item with one-forone replenishment. Management Science 31 (10), 1247–1256.

Healy, K., Schruben, L.W., 1991. Retrospective simulation response optimization. In: Nelson, B.L. , Kelton, D.W., Clark, G.M. (ED.), Proc. 1991 Winter Simulation Conf.,
Institute of Electrical and Electronics Engineers, Piscataway, NJ, 954–957.

Herer, Y.T., Tzur, M., Yucesan, E., 2006. The multilocation transshipment problem. IIE Transactions 38, 185–200.

Homem-de-Mello, T., 2003. Variable-sample methods for stochastic optimization. ACM Transactions on Modeling and Computer Simulation 13 (2), 108–133.

Hopp, W.J., Spearman, M.L., 2000. Factory Physics. McGraw-Hill, New York City, NY.

Hopp, W.J., Zhang, R.Q., Spearman, M.L., 1999. An easily implementable hierarchical heuristic for a two-echelon spare parts distribution system. IIE Transactions 31, 977–988.

Kim, S.-H., Nelson, B.L., 2001. A fully sequential procedure for indifference-zone selection in simulation. ACM Transactions on Modeling and Computer Simulation 11 (3), 251–273.

Kleywegt, A.J., Shapiro, A., Homem-de-Mello, T., 2001. The sample average approxi- mation method for stochastic discrete optimization. SIAM Journal on Optimization 12 (2), 479–502.

K¨ochel, P., Niel¨ander, U., 2005. simulation-based optimisation of multi-echelon inventory systems. International Journal of Production Economics 93-94, 505–513.

Kutanoglu, E., Mahajan, M., 2009. An inventory sharing and allocation method for a multi-location service parts logistics network with time-based service levels. European Journal of Operational Research 194, 728–742.

Little, J.D.C., 1961. A proof of the queueing formula: L = ¸W. Operations Research 9 (3), 383–387.

Mak, H.-Y., Shen, Z.-J.M., 2009. A two-echelon inventory-location problem with service considerations. Naval Research Logistics 56, 730–744.

Muckstadt, J.A., 1973. A model for a multi-item, multi-echelon, multi-indenture inventory system. Management Science 20 (4), 472–481.

Muckstadt, J.A., 2005. Analysis and Algorithms for Service Parts Supply Chains. Springer, New York, NY.

Muckstadt, J.A., Thomas, L.J., 1980. Are multi-echelon inventory methods worth implementing in systems with low-demand-rate items? Management Science 26 (5), 483–494.

Palm, C., 1938. Analysis of the Erlang traffic formula for busy signal arrangements. Ericsson Technics 5, 39–58.

Pasupathy, R., 2010. On choosing parameters in retrospective-approximation algorithms for stochastic root finding and simulation optimization. Operations Research 58 (4), 889–901.

Prakash, P., Deng, G., Converse, M.C., Webster, J.G. , Mahvi, G.M. , Ferris, M.C., 2008. Design optimization of a robust sleeve antenna for hepatic microwave ablation. Physics in Medicine and Biology 53, 1057–1069.

Rustenburg, W.D., Van Houtum, G.J., Zijm, W.H.M., 2001. Spare parts management at complex technology-based organizations: an agenda for research. International Journal of Production Economics 71, 177–193.

Shapiro, A., 1991. Asymptotic analysis of stochastic programs. Annals of Operations Research 30, 169–186.

Shapiro, A., Homem-de-Mello, T., 2000. On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs. SIAM Journal on Optimization 11 (1), 70–86.

Shapiro, A., 2004. Monte Carlo sampling methods. Ruszczynski, A., Shapiro, A. (ED.), Stochastic Programming. Handbooks in Operations Research and Management Science. Elsevier, Amsterdam, 353–426.

Sherbrooke, C.C., 1968. METRIC: a multi-echelon technique for recoverable item control. Operations Research 16 (1), 122–141.

Sherbrooke, C.C., 1986. VARI-METRIC: improved approximations for multi-indenture, multi-echelon availability models. Operations Research 34 (2), 311–319.

Sherbrooke, C.C., 1992. Optimal Inventory Modeling of Systems. Wiley, New York, NY.

Verweij, B., Ahmed, S., Kleywegt, A., Nemhauser, G., Shapiro, A., 2003. The sample average approximation method applied to stochastic vehicle routing problems: A computational study. Computational Optimization and Applications 24, 289–333.

Wang, W., Ahmed, S., 2008. Sample average approximation of expected value con- strained stochastic programs. Operations Research Letters 36, 515–519.

Wang, Y., Cohen, M.A., Zheng, Y.-S., 2000. A two-echelon repairable inventory system with stocking-center-dependent depot replenishment lead times. Management Science 46 (11), 1441–1453.

Wong, H., Kranenburg, B., van Houtum G.-J., Cattrysse, D., 2007. Efficient heuristics for two-echelon spare parts inventory systems with an aggregate mean waiting time constraint per local warehouse. OR Spectrum 29, 699–722.