本研究旨在探討單一製造商與單一零售商的通路結構下，探討有限規劃週期內，產品具有指數損耗特性之聯合訂購與定價決策。首先，我們考量到價格對於需求的影響，因此，將需求假設成與價格相關的函數，以線性模組的方式納入模式中思考，並在採購成本會隨著時間和數量變化的機制下，建構以零售商利潤極大化的角度出發之利潤函數，且分析零售商之利潤函數是否會因為銷售價格因子的考量以及不同的採購時點與不同的採購數量而有所改變，進而釐定定價與訂購週期之聯合決策的問題。
全文在固定的計畫水平期間內考量，同時針對三種情況討論相關的聯合定價與訂購週期決策，分別是允許缺貨、不允許缺貨且完全缺貨待補，和不允許缺貨且部分缺貨待。其中最後者的比例為一和時間相關的函數。在以上提及的三種範疇下，分別先證明其利潤函數具有凹函數特性，更根據這些特性，提供一套求解程序，以決定最適的產品銷售售價、最適存貨補充週期和訂購數量，訂定最佳的存貨補充策略。

The objective of this study is to develop a deterministic inventory model where demand decreases linearly with price, planning horizon is assumed to be finite and the replenishment cycles are constant.Unlike traditional inventory models that seek to minimize cost, the objective of this study is to maximize profit. The net profit maximizing ordering policy and price are determined when purchase cost is time and quantity-dependent.
We consider the cases that do not allow shortages and the ones that allow shortages.
The shortages are completely backlogging and partial backlogging which ensures that the rate of backlogged demand increases as the waiting time to the following replenishment point decreases.

[1] 曹亮吉（2001）,微積分基本要義,三民書局
[2] Abad, P. L.,（1988）,“Determining optimal selling price and lot size when the supplier offers all-unit quantity discounts.”, Decision Sciences, Vol.19, No.3, pp.622-634.
[3] Abad, P. L.,（1988）,“Joint price and lot size determination when supplier offers incremental quantity discount.” Journal of the Operational Research Society, 39, pp.603-607.
[4] Abad, P. L.,（1996）,“Optimal pricing and lot-sizing under conditions of perishability and partial backordering.”, Management Science, Vol. 42, Issue 8, pp.1093-1104.
[5] Aggarwall, S. P. and C. K. Jaggi,（1989）,“Ordering policy for decaying inventory.” International Journal of Systems Science, 20, pp. 151-155.
[6] Bhunia, A. K. and M. Maiti,（1999）,“An inventory model of deteriorating items with lot-size dependent replenishment cost and a linear trend in demand.”, Applied Mathematical Modelling, 23, pp.301-308.
[7] Chang, H. J. and C. Y. Dye,（1999）,“An EOQ model for deterioration items with time varying demand and partial backlogging.” Journal of the Operational Research Society, Vol.50, pp. 1176-1182.
[8] Chen, F., A. Federgruen and Y. S. Zheng,（2001）,“Coordination mechanisms for a distribution system with one supplier and multiple retailers.”, Management Science, Vol.47, No.5, pp.693-708.
[9] Cohen, M. A.,（1977）,“Joint pricing and ordering policy for exponentially decaying inventory with know demand.”, Naval Research Logistics Quarterly, Vol. 24, pp. 257-268.
[10] Covert, R. P. and G.. C. Philip,（1973）,“An EOQ model for items with Weibull distribution deterioration.” AIIE Transactions, 5, pp. 323-326.
[11] Crowther, John F.,（1964）,“Rationale for Quantity Discounts.”, Harvard Business Review, Vol. 42, No. 2, pp.121-127.
[12] Eilon, S. and R. V. Mallya,（1966）,“Issuing and Pricing Policies for Semi-perishables.”, Proceedings of 4th International Conference on Operational Research, Wiley-Interscience.
[13] Emmons, H. and S. M. Gilbert,（1998）,“Note.The role of returns policies in pricing and inventory decisions.”, Management Science, Vol. 44, No.2, pp.276-283.
[14] Ghare, P. M. and G. F. Schrader,（1963）,“A model for an exponentially decaying inventory.”, Journal of Industrial Engineering, Vol. 14, pp. 238-243.
[15] Giri B. C., K. S. Chakrabarty and K. S. Chaudhuri,（2000）,“A note on a lot sizing heuristic for deteriorating items with time-varying demands and shortages.” Computers and Operations Research, 27, pp.495-505.
[16] Goyal S. K., D. Morin and F. Nebebe,（1992）,“The finite horizon trended inventory replenishment problem with shortages.” Journal of Operational Research Society; 43, pp.1173-1178.
[17] Goyal, S. K. and B. C. Giri,（2001）,“Recent trends in modeling of deteriorating inventory.”, European Journal of Operational Research, Vol.134, Issue: 1, pp. 1-16.
[18] Hariga, M. A.,（1994）,“Economic analysis of dynamic inventory models with non-stationary costs and demand.” International Journal of Production Economics, , 35, pp.255-266.
[19] Hariga M. and A. Al-Alyan,（1997）,“A lot sizing heuristic for deteriorating items with shortages in growing and declining markets.” Computers and Operations Research, 24, pp.1075-1083.
[20] Hollier, R. H. and K. L. Mak,（1983）,“Inventory replenishment policies for deteriorating items in a declining market.” International Journal of Production Research, 21, pp.813-826.
[21] Kang, S. and I. Kim,（1983）, “A study on the price and production level of the deteriorating inventory system.” International Journal of Production Research, 21, pp.449-460
[22] Kim, J. S., H. Hwang and S. W. Shinn,（1995）, “An optimal credit policy to increase supplier''s profits with price-dependent demand functions.”, Production Planning and Control, Vol. 6, Issue 1, p45-50.
[23] Monahan, J. P.,（1984）,“A quantity discount pricing model to increase vendor profits.”, Management Science, Vol.30, No.6, pp.720-726.
[24] Munson, C. L. and M. J. Rosenblatt,（2001）,“Coordinating a three- level supply chain with quantity discounts.”, IIE Transactions, Vol.33, No.5, pp.371-384.
[25] Nahmias, S.,（1978）,“Perishable inventory theory：a review.”, Operations Research, Vol. 30, No. 4, pp. 680-708.
[26] Padmanabhan, G. and P. Vrat,（1995）,“EOQ models for perishable items under stock dependent selling rate.” European Journal of Operational Research, Vol. 86, pp.281-292.
[27] Papachristos, S. and K. Skouri,（2000）,“An optimal replenishment policy for deteriorating items with time-varying demand and partial-exponential type-backlogging.”, Operation Research Letters, 27, pp.175-184.
[28] Papachristos, S. and K. Skouri,（2002）,“A continuous review inventory model, with deteriorating items, time-varying demand, linear replenishment cost, partially time-varying backlogging.”, Applied Mathematical Modelling, Volume: 26, Issue: 5, pp.603-617.
[29] Papachristos, S. and K. Skouri,（2003）,“An inventory model with deteriorating items, quantity discount, pricing and time-dependent partial backlogging”, International Journal of Production Economics, 83, pp.247-256.
[30] Park, K. S.,（1982）,“Inventory models with partial backorders.”, International Journal of Systems Science, Vol. 13, pp.1313-1317.
[31] Pekelman, D.,（1974）,“Simultaneous price-production decisions.”, Operations Research, Vol.22, pp.788-794.
[32] Philip, G. C.,（1974）,“A generalized EOQ model for items with Weibull distribution.” AIIE Transactions, 6, pp. 159-162.
[33] Raafat, F.,（1991）,“Survey of literature on continuously deteriorating inventory models.”, Journal of the Operational Research Society, Vol.42, No.1, pp.27-37.
[34] Shah, Y. K.,（1977）,“An order-level lot-size inventory model for deteriorating items.”, AIIE Transactions, Vol.9, pp.108-112.
[35] Shiue, Y. N.,（1990）,“An inventory model for perishable items in a lot size system with quantity discount.” European Journal of Operational Research, 45, pp.260-264.
[36] Teng, Jinn-Tsair; Horng-Jinh Chang, Chung-Yuan Dye and Cheng-Hsing Hung,（2002）,“An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging.”, Operations Research Letters, Volume 30, Issue 6, pp.387-393.
[37] Tersine, R. J. and R. A. Toelle,（1985）,“Lot size determination with quantity discounts.”, Production and Management, Vol.26, No.3, pp.1-23.
[38] Wang, Sheng-Pen （2002）,“An inventory replenishment policy for deteriorating items with shortages and partial backlogging”, Computers and Operations Research, 29, pp 2043-2051
[39] Wee, H. M.,（1992）,“Perishable commodities inventory policy with partial backordering.”, Chung Yuan Journal, Vol. 12, pp. 191-198.
[40] Wee, H. M.,（1995）,“A deterministic lot-size inventory model for deterioration items with shortages and a declining market.” Computers and Industrial Engineering, Vol. 22, pp.345-356.
[41] Wee, H. M.,（1995）,“Joint Pricing and Replenishment Policy for Deteriorating Inventory with Declining Market.”, International Journal of Production Economics, , Vol. 40, Issue 2-3, pp.163-71.
[42] Wee, H. M.,（1997）,“A replenishment policy for items with a price-dependent demand and a varying rate of deterioration.” Production Planning and Control, Vol. 8, Issue 5, pp.494-499.
[43] Wee, H. M.,（1999）,“Deteriorating inventory model with quantity discount, pricing and partial backordering.”, International Journal of Production Economics, Vol.59, pp.511-518.
[44] Weng, Z. K.,（1995）,“Channel coordination and quantity discounts.”, Management Science, Vol. 41, No. 9, pp.1509-1522.
[45] Whitin, T. M.,（1955）,“Inventory control and price theory.”, Management Science, Vol.2, No.1, pp.61-68.
[46] Whitin, T. M.,（1957）,“Theory of Inventory Management”, Princeton University Press, Princeton