本研究探討在連續型檢視之存貨系統(continuous review inventory systems)下，如何整合兩種不同隨機型需求(stochastic demands)之訂單以達成最小化總成本之目標－(1)有簽訂長期合約的顧客訂單－class 1，具有第一優先的供給順序及(2)偶發性需求的顧客訂單－class 2，兩者若有缺貨皆視為銷售損失(lost sale)。在每一次的訂購週期(time between replenishment)中皆須決定該供給多少數量給class 2之訂單，依此方向本研究發展－(s，M，Q)配給(rationing)模式，s及Q為決定何時訂購和須訂購多少量，M為當存貨水準(inventory level)高於此水準時，視兩訂單型態為先到先服務；若降至或低於此水準時，即停止供給class 2之訂單，亦即class 2訂單雖抵達但不供給其需求，視為銷售損失。根據以上所述，本研究首先針對單一產品依序推導「倉庫(warehouse)-顧客(customer)」、「生產(production)-倉庫(warehouse)」二種管理模式的供給問題並推導出最佳訂購量之決策特性，然後再依「倉庫-顧客」管理模式所推導出的決策特性發展多產品(multiple products)之管理模式。接著，根據各模式所推導之決策特性，發展演算法並經由與matlab內建之尋找最佳解的程式做一比較，藉以證實本研究所推導之管理模式之決策特性可以找到最佳解且求解速度更快。

A supplier/warehouse face demands from different kinds of customers: some of them frequently place orders at almost constant rates, but others infrequently place orders of random sizes.To deal with different stochastic demand patterns at the same time, an optimal inventory control model is developed based on a rationing policy. Nonlinear programming models are derived by minimizing inventory costs to find the optimal reorder point, reorder quantity and rationing point. The derived inventory cost models consider a single product as well as multiple products with limiting storage space. On the basis of the properties of the inventory cost models, optimal solution searching algorithms are developed to determine the values of decision variable. Numerical studies provided in this research show that the algorithms perform well and are easy to compute.

Ashayeri, J. and Selen, W. J., (2001). “Order selection optimization in hybrid make-to-order and make-to-stock markets”, Journal of the Operational Research Society, 52(10), 1098-1106.
Balakrishnan, N., Patterson, J. W., and Sridharan, V., (1999). “Robustness of capacity rationing policies”, European Journal of Operational Research, 115(2), 328-338.
Carr, S. and Duenyas, I., (2000). ”Optimal admission control and sequencing in a make-to-stock/make-to-order production system”, Operations Research, 48(5), 709-720.
Chang, S. H., Pai, P. F., Yuan, K. J., Wang, B. C., and Li, R. K., (2003).“Heuristic PAC model for hybrid MTO and MTS production enviroment”, International Journal of Production Economics, 85(3), 347-358.
Deshpande, V., Cohen, M. A., and Donohue, K., (2003). “Threshold inventory rationing policy for service-differentiared demand classes”, Management Science, 49(6), 683-703.
Duran, S., Liu, T., Simchi-Levi, D., and Swann, J. L., (2008). “Policies utilizing tactical inventory for service-differentiated customers”, Operations Research Letters, 36(2), 259-264.
Federgruen, A. and Katalan, Z., (1999). “The impact of adding a make-to-order item to a make-to-stock production system”, Management Science, 45(7), 980-994.
Frank, K. C., Zhang, R. Q., and Duenyas, I., (2003). “Optimal policies for inventory systems with priority demand classes”, Operations Research, 51(6), 993-1002.
Ha, A. Y., (1997). “Inventory rationing in a make-to-stock production system with several demand classes and lost sales”, Management Science, 43(8), 1093-1103.
Ha, A. Y., (1997). “Stock-rationing policy for a make-to-stock production system with two priority classes and backordering”, Naval Research Logistics, 44(5), 457-472.
Kogan, K., Khmelnitsky, E., and Maimon, O., (1998). “Balancing facilities in aggregate production planning:make-to-order and make-to-stock environments”, International Journal of Production Research, 36(9), 2585-2596.
Melchiors, P., (2003). “Restricted time-remembering policies for the inventory rationing problem”, International Journal of Production Economics, 81-82, 461-468.
Melchiors, P., Dekker, R., and Kleijn, M. J., (2000). “Inventory rationing in an (s, Q) inventory model with lost sales and two demand classes”, Journal of the Operational Research Society, 51(1), 111-122.
Moon, I., and Kang, S., (1998). “Rationing policies for some inventory systems”, Journal of the Operational Research Society, 49(5), 509-518.
Nahmias, S. and Demmy, W. S., (1981). “Operating characteristics of an inventory system with rationing”, Management Science, 27(11), 1236-1245.
Rajagopalan, S., (2002). “Make to order or make to stock：Model and application”, Management Science, 48(2), 241-256.
Silver, E. A., Pyke, D. F., and Peterson, R., (1998). “Inventory management and production planning and scheduling(3rd ed.)”, John Wiley & Sons, Inc.
Sobel, M. J. and Zhang, R. Q., (2001). “Inventory policies for systems with stochastic and deterministic demand”, Operations Research, 49(1), 157-162.
Soman, C. A., Van Donk, D. P., and Gaalman, G., (2004). “Combined make-to-order and make-to-stock in a food production system”, International Journal of Production Economics, 90(2), 223-235.
Topkis, D. M., (1968). “Optimal ordering and rationing policies in a nonstationary dynamic inventory model with n demand classes”, Management Science, 15(3), 160-176.
Veinott Jr., A. F., (1965). “Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes”, Operations Research, 13(5), 761-778.
潘昭賢、葉瑞徽譯(民國90)。Hillier/Lieberman(1967)原著。「作業研究(下)，第七版」(Introduction To Operations Research, 7th ed)。滄海書局。