針對飛機票、旅館訂房及流行性商品等具有時效性的商品，多產品的問題日漸普及，消費者可以選擇購買較低廉的商品，或在可接受的範圍內付出較高的金額以換取更好的產品服務水準。業者必須在有限的銷售期間同時考慮到顧客對於商品個別的反應以及比價所造成的影響，以求取更高的收益。因此，針對獨占市場中兩個替代性商品的問題，我們提出了動態價格模式，目的是使得業者的收益最大化。
本研究利用多項羅吉特模式描述顧客的選擇行為，並藉由計算轉換價格的最佳時間序列，使得業者可以在連續時間過程中動態地調整價格。此外，我們利用邊際利潤的概念分析出有效的價格策略以降低運算時間。最後，以一些數據的例子說明模式的進行步驟，並將實際的收益與固定價格模式的結果作一比較。

As multi-product problem becomes more and more common today, customers may substitute their initial choice to a cheaper one they can afford or upgrade with additional payment to get higher service level. In many industries, such as airline seats allocation, hotel rooms or seasonal fashion goods, seller must consider customer reaction from the strategic prices on each product and its interaction to enhance the revenue in a finite sales horizon. Hence, we proposed a dynamic pricing model where a sell offers two substitutable perishable products in the monopolist market. The objective is to maximize the total profit gain from both two products.
In this thesis, we analyze the problem using a multinomial logit model to describe the customer choice behavior. The exact solution is obtained according to the calculated optimal time thresholds. Sellers can dynamically adjust the price policy in the continuous-time process. We also analyze the efficient price strategy based on the marginal expected revenue function. Furthermore, a numerical example is illustrated to show the procedure and the resultant is compared with the fixed price policy.

1. Bassok, Y., Anupindi, R. and Akella, R., “Single-Period Multiproduct Inventory Model with Substitution,” Operations Research, 47 (4), 632-642 (1999).
2. Belobaba, P.P., “Airline yield management: An overview of seat inventory control,” Transportation Science, 21, 63-73 (1987).
3. Bitran, G. and Caldentey, R., “An Overview of Pricing Model for Revenue Management,” Manufacturing & Service Operations Management, 5(3), 203-2009 (2003).
4. Bitran, G., Caldentey, R. and Vial, R., ”Pricing Policies for Perishable Products with Demand Substitution, ” Working Paper, Sloan School of Management, MIT, Cambridge, U.S.A. (2005)
5. Brumelle, S.L. and McGill, J.I., “Airline Seat Allocation with Multiple Nested Fare Classes,” Operations Research, 41 (1), 127-137 (1993).
6. Chen, Y.H., “Dynamic Pricing Model on the Internet Market,” Thesis. Department of Industrial Engineering and Engineering Management National Tsing Hua University, Taiwan, R.O.C (2004).
7. Chun, Y.H., “Optimal Pricing and Ordering Policies for Perishable Commodities,” European Journal of Operational Research, 144, 68-82 (2003)
8. Feng, Y. and Xiao, B., “A Continuous-Time Yield Management Model With Multiple Prices and Reversible Price Changes,” Management Science, 46 (5), 644-657 (2000).
9. Feng, Y. and Xiao, B., “Optimal Policies of Yield Management with Multiple Predetermined Prices,” Operations Research, 48 (2), 332-343 (2000).
10. Feng, Y. and Gallego, G., “Perishable Asset Revenue Management with Markovian Time Dependent Demand Intensities,” Management Science, 46 (7), 941-956 (2000).
11. Feng, Y. and Xiao, B., “A Dynamic Airline Seat Inventory Control Model and Its Optimal Policy,” Operations Research 49 (6), 938-949 (2001).
12. Gallego, G. and van Ryzin, G.., “Optimal Dynamic Pricing of Inventories with Stochastic Demand over Finite Horizons,” Management Science, 40, 999-1020 (1994)
13. Lee, T.C. and Hersh, M., “A Model for Dynamic Airline Seat Inventory Control with Multiple Seat Bookings,” Transportation Science, 27 (3), 252-265 (1993).
14. Li, F. and Lin, K.Y., “Optimal Dynamic Pricing for Two Perishable and Substitutable Products,” Working Paper, Grado Department of Industrial and System Engineering, Virginia Tech., (2004).
15. Li, M.Z.F., “Pricing Non-storable Perishable Goods by Using a Purchase Restriction with an Application to Airline Fare Pricing,” European Journal of Operational Research, 134, 631-347 (2001).
16. Lin, K.Y., “A Sequential Dynamic Pricing Model and Its Applications,” Naval Research Logistics, 51 (4), 501-521 (2004).
17. Lin, K.Y. and Li, F., “Optimal Dynamic Pricing for a Line of Substitutable,” Working Paper, Grado Department of Industrial and System Engineering, Virginia Tech., (2004).
18. Littlewood, K., “Forecasting and Control of Passengers,” 12th AGIFORS Symposium Proceedings, Nathanya, Israel 95-128 (1972).
19. Weatherford, L.R. and Bodily, S.E. “A Taxonomy and Research Overview of Perishable-Asset Revenue Management: Yield Management, Overbooking, and Pricing,” Operations Research 40(5), 831-844 (1992).
20. Mahajan, S. and van Ryzin, G.J., Quantitative Models for Supply Chain Management, chapter 17, 505-506 (2000).
21. McGill, J.I. and van Ryzin, G.., “Revenue Management: Research Overview and Prospects,” Transportation Science 33 (2), 233-255 (1999).
22. Monahan, G.E., Zhao, W. and Petruzzi, N.C., “The Dynamic Pricing Problem from a Newsvendor's Perspective,” Manufacturing & Service Operations Management, 6 (1), 73-91 (2004).
23. Sawaki, K., “Optimal Policies in Continuous Time Inventory Control Models with Limited Supply,” Computers and Mathematics with Applications 46, 1139-1145 (2003).
24. Smith, B., Leimkuhler, J.F. and Darrow R.M., “Yield Managemnet at American Airlines,” Interfaces, 22(1), 8-31 (1992).
25. Subramanian, J., Stidham S. Jr. and Lautenbacher, C.,”Airline yield management with overbooking, cancellations and no-shows,” Transportation Science, 33, 147-167 (1999).
26. Talluti, K. and van Ryzin, G.., “Revenue Management under a General Discrete Choice Model of Consumer Behavior,” Management Science, 50(1), 15-33 (2004).
27. Van Ryzin, G. and Mahajan S., “On the Relationship between Inventory Costs and Variety Benefits in Retail Assortments.” Management Science, 45(11), 1496-1509 (1999).
28. Zhao, W. and Zheng, Y.S. “Optimal Dynamic Pricing for Perishable Assets with Nonhomogeneous Demand,” Management Science, 46 (3), 375-388 (2000).
29. Zhao, W. and Zheng, Y.S., “A Dynamic Model for Airline Seat Allocation with Passenger Diversion and No-Shows,” Transportation Science, 35 (1), 80-98 (2001).