經濟生產批量（EPQ）模型是以經濟訂購量（EOQ）模型為基礎所延伸出的
存貨模型，後續也有許多學者將原先條件放寬而做進一步的分析。
本篇論文分析了需求量會隨時間改變下的退化性商品庫存模式，主要是以指數型需求曲線與缺貨後補狀況為其基礎模型，並分別在物品的生產率是需求率的隨機倍數關係，以及再訂購點也為隨機變數的狀況與條件下，我們分析其最佳的存貨策略，討論一種有效的方式來獲得最優的再訂購點或是最佳的存貨周期，並且使整體的每單位平均總成本必須達到最小。在此，隨機變數假設為來自連續型的均勻分配 (uniform distribution)。
本篇論文也會提供一些數值例子來說明所提出的存貨模型，並分析在某些參數改變下，最佳存貨週期、最佳再訂購點與期望平均總成本，會產生如何的相對變化。

The Economic Production Quantity (EPQ) model is an inventory model extended from the traditional Economic Order Quantity (EOQ) model. Many scholars had subsequently relaxed the original conditions for further analyses.
This paper analyzes the inventory model with time-varying demand, and considers the power demand curve and backlogged shortage condition. The production rate is stochastically proportional to demand rate, while the random variable is subject to a uniform distribution of continuous type. In addition, the reorder point is stochastic, while the random variable is also subject to an uniform distribution of continuous type. Under above conditions, the optimal inventory policy is studied and an effective way is provided to obtain the best inventory cycle time, the best reorder point, and the minimum total cost per unit per unit time.
This paper will also provide some numerical examples to illustrate the proposed inventory model, and analyze how the inventory cycle time, optimal reordering point, and the expected average total cost will be changed under the changes of certain parameter.

參考文獻

[1] Barbosa, L.C., Friedman, M., 1978. Deterministic inventory lot size models—a general root law. Manage. Sci. 24 (8), 819–826.

[2] Bose, S., Goswami, A., Chauddhuri, K.S., 1995. An EOQ model for deteriorating items with linear time-dependent demand rate and shortages under inflation and time discounting. J. Oper. Res. Soc. 46 (6), 771–782.

[3] Chakrabarti, T., Chaudhuri, K.S., 1997. An EOQ model for deteriorating items with a linear trend in demand and shortages in all cycles. Int. J. Prod. Econ. 49 (3), 205–213.

[4] Datta, T.K., Pal, A.K., 1988. Order level inventory system with power demand pattern for items with variable rate of deterioration. Indian J. Pure Appl. Math. 19 (11), 1043–1053.

[5] Dave, U., Patel, R.K., 1981. (T,Si) Policy inventory model for deteriorating items with time proportional demand. J. Oper. Res. Soc. 32 (2), 137–142.

[6] Donalson, W.A., 1977. Inventory replenishment policy for a linear trend in demand: an analytical solution. Oper. Res. Q. 28 (3), 663–670.

[7] Dye, C-Y., 2004. A note on “an EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern”. Int. J. Inf. Manage. Sci. 15 (2), 81–84.
[8] E. W. Taft, 1918. The Most Economical Production Lot, The Iron Age, Vol. 101, 1918, pp.1410-1412.

[9] Goel, V.P., Aggarwal, S. P., 1981. Order level inventory system with power demand pattern for deteriorating items. In: Proceedings of the All India Seminar on Operational Research and Decision Making, University of Delhi, New Delhi, 19–34.

[10] Goswani, A., Chaudhuri, K.S., 1991. EOQ Model for an inventory with a linear trend in demand and finite rate of replenishment considering shortages. Int. J. Syst. Sci. 22 (1), 181–187.

[11] Goyal, S.K., Giri, B.C., 2003. A simple rule for determining replenishment intervals of an inventory item with linear decreasing demand rate. Int. J. Prod. Econ. 83 (2), 139–142.

[12] Harris, F.W., 1913. How many parts to make at once. Factory, The Magazine of Management 10 (2), 135–136. (152).

[13] Hsueh, C-F., 2011. An inventory control model with consideration of remanufacturing and product life cycle. Int. J. Prod. Econ. 133 (2), 645–652.

[14] Lee, W-C., Wu, J-W., 2002. An EOQ Model for Items with Weibull distributed deterioration, shortages and power demand pattern. Int. J. Inf. Manage. Sci. 13 (2), 19–34.

[15] Maihami, R., Kamalabadi, I.N., 2012. Joint pricing and inventory control for noninstantaneous deteriorating items with partial backlogging and time and price dependent demand. Int. J. Prod. Econ. 136 (1), 116–122.

[16] Mishra, V.K., Singh, L.S., 2010. Deteriorating inventory model with time dependent demand and partial backlogging. Appl. Math. Sci. 4 (72), 3611–3619.

[17] Mishra, S.S., Singh, P.K., 2013. Partial backlogging EOQ model for queued customers with power demand and quadratic deterioration: computational approach. Am. J. Oper. Res. 3 (2), 13–27.

[18] Mitra, A., Cox, J.F., Jesse, R.R., 1984. A note on deteriorating order quantities which a linear trend in demand. J. Oper. Res. Soc. 35 (2), 141–144.

[19] Naddor, E., 1966. Inventory Systems. John Wiley and Sons, New York.

[20] Omar, M., Yeo, I., 2009. A model for a production–repair system under a timevarying demand process. Int. J. Prod. Econ. 119 (1), 17–23.

[21] Pando, V., García-Laguna, J., San-José, L.A., 2012. Optimal policy for profit maximising in an EOQ model under non-linear holding cost and stock-dependent demand rate. Int. J. Syst. Sci. 43 (11), 2160–2171.

[22] Pando, V., San-José, L.A., García-Laguna, J., Sicilia, J., 2013. An economic lot-size model with non-linear holding cost hinging on time and quantity. Int. J. Prod. Econ. 145 (1), 294–303.

[23] Rajeswari, N., Vanjikkodi, T., 2011. Deteriorating inventory model with power demand and partial backlogging. Int. J. Math. Arch. 2 (9), 1495–1501.

[24] Rajeswari, N., Vanjikkodi, T., 2012. An inventory model for items with two parameter Weibull distribution deterioration and backlogging. Am. J. Oper. Res. 2 (2), 247–252.

[25] Resh, M., Friedman, M., Barbosa, L.C., 1976. On a general solution of the deterministic lot size problem with time-proportional demand. Oper. Res. 24 (4), 718–725.

[26] Ritchie, E., 1984. The EOQ for linear increasing demand: a simple optimal solution. J. Oper. Res. Soc. 35 (10), 949–952.

[27] Sakaguchi, M., 2009. Inventory model for an inventory system with time-varying demand rate. Int. J. Prod. Econ. 122 (1), 269–275.

[28] Sicilia, J., Febles-Acosta, J., González-De la Rosa, M., 2012. Deterministic inventory systems with power demand pattern. Asia-Pac. J. Oper. Res. 29 (5). (article no. 1250025 (28 pages).

[29] Singh, T.J., Singh, S.R., Dutt, R., 2009. An EOQ model for perishable items with power demand and partial backordering. Int. J. Oper. Quant. Manage. 15 (1), 65–72.

[30] Silver, E.A., 1979. A simple inventory replenishment decision rule for a linear trend in demand. J. Oper. Res. Soc. 30 (1), 71–75.

[31] Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, third ed.. John Wiley Sons, Inc., New York.

[32] Stoer, J., Bulirsch, R., 2002. Introduction to Numerical Analysis, third ed.. Springer, New York.

[33] Wen-Yang, L., Chih-Hung, T., Rong-Kwei, L., 2002. Exact solution of inventory replenishment policy for a linear trend in demand—two-equation model. Int. J. Prod. Econ. 76 (2), 111–120

[34] Wu, K-S., 2002. Deterministic inventory model for items with time varying demand, Weibull distribution deterioration and shortages. Yugoslav J. Operations Res. 12 (1), 61–71.

[35] Yang, J., Zhao, G.Q., Rand, G.K., 2004. An eclectic approach for replenishment with non-linear decreasing demand. Int. J. Prod. Econ. 92 (2), 125–131.

[36] Zhao, G.Q., Yang, J., Rand, G.K., 2001. Heuristics for replenishment with linear decreasing demand. Int. J. Prod. Econ. 69 (3), 339–345.
[37] Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw Hill, Boston.

[38] Joaquin Sicilia, Manuel Gonzales-De-la-Rosa, Jaime Febles-Acosta, David Alcaide-Lopez-de-Pablo. Optimal policy for an inventory system with power demand backlogged shortages and production rate proportional to demand rate.