網際網路的盛行帶動了製造業邁向另外一個經營模式。最近幾年，消費者除了到實體店面挑選物品，他們可以在電腦前選購自己最喜愛的商品。為了解決零售商的存貨問題，單期存貨問題是目前廣為所探討的議題之一。傳統的單期存貨問題或報童問題是在期望利潤最大化的目標及需求服從某特定分配下找出最佳的訂購量。
單期商品的其中一個特性為允許一個銷售期間。因此，管理者必須在銷售期間尚未開始前先決定產品數量。然而，根據很多國家的法律，若發現商品有瑕疵，或在螢目上所看到的商品跟實際拿到的商品的尺寸及顏色有一些差異，消費者可以在法律所規定的時間內向零售商退貨。這些被退貨的商品經由維修後重新包裝可以進行再銷售。另外，由於行銷部門採取一些廣告活動來吸引潛在的客戶，當期的需求量有可能會過大。因此，為了滿足願意等待的消費者，管理者必須考慮是否要進行緊急補貨。
本研究擬將延伸傳統的單期存貨問題，考量存貨政策，緊急補貨，及廣告效果，在期望利潤極大化的目標及需求服從均勻及常態分配下，決定最佳的存貨量及廣告成本。為顯現模式的實用性，本研究除了提供一個模擬案例，對於各個廣告後的模式進行敏感度分析。

Since the emerging of internet, online catalogues have contributed substantial profit in total sales of manufacturing industries. One of the topics that have been studied in this area is the single-period problem. The classical single-period problem (SPP) or the so-called newsboy/newsvendor problem is to find a product’s order quantity that maximizes the expected profit under probabilistic demand.
One of the characteristics of fashionable products is that they can only be sold within a single selling period. Managers have to predict the optimal quantity of products in advance. The problem can be more complicated for these style products since returns may also be used to satisfy new demand, occurred before the end of selling period. In other case, it is unavoidable that there is a possibility of shortage due to extremely high demand during the period. Here, vendor provides emergency supply option to satisfy a proportion of customers who are willing to wait. Since there are some vendors selling these products within an area, consequently, the rest of customers, unwilling to wait to order in one store, may seek the products in others.
In this research, we extend the classical single-period problem to determine the optimal order quantity and advertising expenditure by incorporating return policies, emergency supply options, and the effect of advertising under the objective of maximizing the expected profit. We provide solutions to the problem under uniform and normal demand distributions. A numerical example is used to illustrate the model, and the associated sensitivity analyses in terms of some important parameters are also made.

Abdel-Malek, L.L., Montanari, R., On the multi-product newsboy problem with two constraints. Computer & Operations Research, 32, 2095-2116, 2005.

Alfares, H.K., Elmorra, H.H., The distribution-free newsboy problem: Extensions to the shortage penalty case. International Journal of production Economics, 93-94, 465-477, 2005.

Alferdsson, P., Verrijdt, J., Modeling emergency supply flexibility in a two-echelon inventory system. Management Science, 45(10), 1416-1431, 1999.

Axster, S., A heuristic for triggering emergency orders in an inventory system. European Journal of Operational Research, 176, 880-891, 2007.

Cachon, G., Terwiesch, C., Matching supply with demand: an introduction to operations management, 1st ed., McGraw-Hill, 2006.

Chiang, C., Optimal replenishment for a periodic review inventory system with two supply modes. European Journal of Operational Research, 149, 229-244, 2003.

Chopra, S., Meindl, P., Supply chain management: strategy, planning and operations, 2rd ed., Prentice Hall, 2004.

D’Souza, G., Allaway, A., An empirical investigation of the advertising spending decisions of a multi-product retailer. Journal of Retailing, 71(3), 279-296, 1995.

Dominey, M.J.G., Hill, R.M., performance of approximations for compound Poisson distributed demand in the newsboy problem. International Journal of production Economics, 92, 145-155, 2004.

East, R., The effect of advertising and display: assessing the evidence, Springer, 2003.

Emmons, H., Gilbert, S.M., Note: the role of returns policies in pricing and inventory decisions for catalogue goods. Management Science, 44, 2, 1998.

Erlebacher, S.J., Optimal and heuristic solutions for the multi-item newsvendor problem with a single constraint. Production and Operations Management, 27, 537-553, 1999

Farris, P.W., Reibstein, D.J., How prices, expenditures, and profits are linked. Harvard Business Review, 57(6), 173-184, 1979

Fisher, M.L., Raman, A., Reducing the cost of demand uncertainty through accurate response to early sales. Operations Research, 44(1), 87-99, 1996.

Gallego, G., Moon, I., The distribution free newsboy problem: review and extensions. Journal of operational Research Society, 44(8), 825-834, 1993.

Ha, L., Litman, B.R., Does advertising clutter have diminishing and negative returns. Journal of Advertising, 26(1), 31-42, 1997.

Hadley, G.., Whiting, T.M., Analysis of Inventory Systems, Prentice Hall, 1963.

Huang, Z., Li, S.X., Co-op advertising models in manufacturer-retailer supply chains: a game theory approach. European Journal of Operational Research, 135, 527-544, 2001.

Johansen, S.G., Thorstenson, A., An inventory model with Poisson demands and emergency orders. International Journal of Production Economics, 56, 275-289, 1998.

Khouja, M., The newsboy problem under progressive multiple discounts. The European Journal of Operational Research, 84, 458-466, 1995.

Khouja, M., The single-period (news-vendor) problem: literature review and suggestions for future research. Omega, 27, 537-553, 1999.

Khouja, M., Optimal ordering, discounting, and pricing in the single-period problem. International Journal of production Economics, 65, 201-216, 2000.

Khouja, M., Robbins, S.S., Linking advertising and quantity decisions in the single-period inventory model. International Journal of Production Economics, 86, 93-105, 2003.

Kotler, P., Ang, S.H., Leong, S.M., Tan, C.T., Marketing Management: An Asian Perspective, 8rd ed., Prentice Hall, 1999.

Krajewsky, L.J. and Ritzman, L.P., Operations management: strategy and analysis, 6rd ed., Prentice Hall, 2002.

Kurawala, A.A., Matsuo, H., Forecasting and inventory management of short life-cycle products. Operations Research, 44(1), 131-150.

Lau, H.S., The newsboy problem under alternative optimization objectives. Journal of Operational Research Society, 31, 525-535, 1980.

Lau, A., Lau, H., The newsboy problem with price-dependent demand distribution. IEEE Transactions, 20, 168-175, 1988

Lau, H.S., Simple formulas for the unexpected costs in the newsboy problem: an educational note. European Journal of Operational Research, 100, 557-561, 1997.

Lau, H.S., Lau, A.H.L., Reordering Strategies for a newsboy-type product. European Journal of Operational Research, 103, 557-572, 1997

Lau, H.S., Lau, H.L., Manufacturer’s pricing strategy and return policy for a single-period commodity. European Journal of Operational Research, 116, 291-304, 1999.

Li, J., Lau, H., Lau, A.H., A two-product newsboy problem with satisfying objective and independent exponential demands. IEE Transactions, 23, 29-39, 1991.

Li, S.X., Huang, Z., Zhu, J., Chau, P.Y.K., Cooperative advertising, game theory, and manufacturer-retailer supply chains. Omega, 30, 347-357, 2002.

Meyer, H., Many happy returns. The Journal of Business Strategy, 20(4), 27-31.

Mohebbi, E., Posner, M.J.M., A lost-sales continuous-review inventory system with emergency ordering. International Journal of Production Economics, 58, 93-112, 1999

Moinzadeh, K., Nahmias, S., A continuous review model for an inventory system with two supply modes. Management Science, 34, 761–773, 1988.

Mostard, J., Koster, R.d., Teunter, R., The distribution-free newsboy problem with resalable returns. International Journal of Production Economics, 97, 329-342, 2005.

Mostard, J., Teunter, R., The newsboy problem with resalable returns: A single period model and case study. European Journal of Operational Research, 169, 81-96, 2006.

Mukhopadhyay, S.K., Setoputro, R., Optimal return policy and modular design for build-to-order products. Journal of Operations Management, 23, 496-506, 2005.

Pasternack, B.A., Optimal pricing and return policies for perishable commodities. Marketing Science, 4(2), 1985.

Petruzzi, N.C., Dada, M., Pricing and the newsvendor problem: A review with extensions. Operations Research, 47(2), 183-194, 1998.

Silver, E. A., Pyke, D.F., Peterson, R., Inventory management and production: planning and scheduling, 3rd ed., John Wiley & Sons, 1998.

Vlachos, D., Rommert, D., Return handling options and order quantities for single period products. European Journal of Operational Research, 151, 38-52, 2003.

Weatherford, L.R., Pfeifer, P.E., The economic value of using advanced booking of orders. Omega, 44, 825-834, 1994.

Yue, J., Austin, J., Wang, M.C., Huang, Z., Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount. European Journal of Operational Research, 168, 65-85, 2006.