此篇論文研究了閉環式供應鏈的四種模型，包含分散式的 Stackelberg game，分別由製造商、零售商、獨立的回收商負責從事回收活動，以及集中式的中心系統整合模式，將製造商和零售商視為一體，賣產品給消費者，並向消費者回收舊品，接著進行再製。在這兩種不同競爭模式的賽局理論基礎下，我們假設製造商生產率大於消費者對於產品的需求率，發展各別的存貨系統，使用數學模式，將閉環式供應鏈的四種模型描述成利潤最大化的問題，並且利用演算法來求得最佳解/均衡解，進而分析與比較各個最佳化後的模式。最後，提供了一些數值範例來闡述問題的特性，而得到了這樣的結論：集中式的再製系統在所有模型之中有最好的績效，接著接續為零售商做回收、製造商做回收，最後則是第三方負責回收。

This paper investigates four models for the closed-loop supply chain including the decentralized Stackelberg game and centrally coordinated system model. There are different roles which are responsible for collecting the used products from the consumers: the manufacturer, the retailer and the independent third party. For the centralized model, we view the manufacturer and the retailer as one central planner who sells products to the consumers and buys the used products back from them, and then remanufactures those returned products. Under the basis of these two different game theories, we assume that the manufacturer’s production rate is greater than the consumers’ demand rate for the products and develop the inventory systems of the four closed-loop supply chain models and describe them as profit maximization problems by mathematical formulations. The problems are solved by developing algorithms to find the optimal/equilibrium solutions, and the analytical results for the optimized closed-loop supply chain structures are presented. At last, some numerical examples are provided to illustrate the features of our models. Moreover, we can get such conclusions: centrally coordinated system collection is the best profitable model which is better than retailer collection, manufacturer collection, and third-party collection sequentially.

1. Amin, S.H., Zhang, G., 2012. An integrated model for closed–loop supply chain configuration and supplier selection: Multi–objective approach. Expert Systems with Applications 39, 6782–6791.
2. Benton, W.C., 2000. Supply–chain management for recoverable manufacturing systems. Interface 30, 125–142.
3. Bras, B., McIntosh, M., 1999. Product, process and Organizational design for remanufacture – an overview of research. Robotics and Computer Integrated Manufacturing 15, 167–178.
4. Chung, S.L., Wee, H.M., Yang, P.C., 2008. Optimal policy for a closed–loop supply chain inventory system with remanufacturing. Mathematical and Computer Modelling 48, 867–881.
5. Sheu C., Yen H.J., Chae B., 2006. Determinants of supplier–retailer collaboration: evidence from an international study. International Journal of Operations & Production Management 26, 24–49.
6. Daniel, V., Guide, J.R., Wassenhove, V., Luk N., 2006. Closed–Loop Supply Chains: An Introduction to the Feature Issue (Part 1). Production and Operations Management 15 (3), 345–350.
7. Daniel, V., Guide, J.R., Wassenhove, V., Luk N., 2006. Closed–Loop Supply Chains: An Introduction to the Feature Issue (Part 2). Production and Operations Management 15 (4), 471–472.
8. Dekker, R., Fleishmann, M., Inderfurth, K., Wassenhove, V., Luk N., 2004. Reverse Logistics: Quantitative Models for Closed–Loop Supply Chains.
9. Deng, Y.H., 2009. An optimal replenishment policy for two–echelon inventory system with remanufacturing.
10. Fleischmann, M., Bloemhof–Fuwaard, J.M., Dekker, R., Laan, Erwin A., Van Nunen, J.A.E.E., Wassenhove, V., Luk N., 1997. Quantitative models for reverse logistics: A review. European Journal of Operational Research 103 (1), 1–17.
11. Guide, D., Harrison, T., Wassenhove, L.V., 2003. The challenge of Closed-Loop supply Chains. Interface 33 (6), 3–6.
12. Guide, V.D.R., Jayaraman, V., Srivastava, R., Benton, W.C., 2000. Supply–chain management for recoverable manufacturing systems. Interfaces 30 (3), 125–142.
13. Hatcher, G.D., Ijomah, W.L., Windmill, J.F.C., 2011. Design for remanufacture: a literature review and future research needs. Journal of Cleaner Production 19, 2004–2014.
14. Hong, I.H., Yeh, J.S., 2012. Modeling closed–loop supply chains in the electronics industry: A retailer collection application. Transportation Research Part E 48 (4), 817–829.
15. Koh, S.G., Hwang, H., Sohn, K.I., Ko, C.S, 2002. An optimal ordering and recovery policy for reusable items. Computers & Industrial Engineering 43 (1–2), 59–73.
16. Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E, 1998. Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM Journal of Optimization 9 (1), 112–147.
17. Lin, C.F., 2010. Optimal policies for the closed-loop supply chain inventory systems with product remanufacturing under price–sensitive demand consideration.
18. Luther, L., 2008. Managing Electronic Waste: An Analysis of State E–Waste Legislation. CRS Report for Congress.
19. Mohsen, S.S., Mohammad, R.A.J., 2009. Optimizing shipment, ordering and pricing policies in a two–stage supply chain with price-sensitive demand. Transportation Research Part E 45, 564–571.
20. Nahmias, S., Rivera, H., 1979. A deterministic model for a repairable item inventory system with a finite repair rate. International Journal of Production Research 17 (3), 215–221.
21. Nie, J., Huang, Z., Zhao, Y., Shi, Y., 2013. Collective Recycling Responsibility in Closed–Loop Fashion Supply Chains with a Third Party: Financial Sharing or Physical Sharing? Mathematical Problems in Engineering 2013.
22. Olugu, E.U., Wong, K.Y. 2012. An expert fuzzy rule–based system for closed–loop supply chain performance assessment in the automotive industry. Expert Systems with Applications 39, 375–384.
23. Qina, Y., Tanga, H., Guoa, C., 2007. Channel coordination and volume discounts with price–sensitive demand. International Journal of Production Economics 105, 43–53.
24. Richter, K., Dobos, I., 1999. Analysis of the EOQ repair and waste disposal problem with integer setup numbers. International Journal of Production Economics 59, 463–467.
25. Sakon, W., Bordin, R., 2011. The determination of high cost and low cost spare parts by using the comparison between EOQ model and lot–for–lot inventory model: A case study of slow moving item. IEEE, 115–121.
26. Savaskan, R.C., Bhattacharya, S., Wassenhove, L.V., 2004. Close-loop supply chain models with product remanufacturing. Management Science 50 (2), 239–252.
27. Savaskan, R.C., Wassenhove, L.V., 2006. Reverse Channel Design: The Case of Competing Retailers. Management Science 52 (1), 1–14.
28. Schrady, D.A., 1967. A deterministic inventory model for reparable items. Naval Research Logistics 14 (3), 391–398.
29. Schwarz, L.B., 1972. Economic order quantities for products with finite demand horizons. AIIE Transactions 4 (3).
30. Seegrid Corporation. 2008. GENCO Supply Chain Solutions. Seegrid Corporation.
31. Shumon, R.H., Kazi, A.U.Z., Azizur, R., 2011. Prospects of Remanufacturing, Bangladesh perspective. International Journal of Industrial Engineering: Theory, Applications and Practice 18 (5), 254–259.
32. Teierry, M., Salomon, M., Nunen, J.V., Wassenhove, L.V., 1995. Strategic issues in product recovery management. California Management Review 37 (2), 114–135.
33. Teunter, R.H., 2001. Economic ordering quantities for recoverable item inventory systems. Naval Research Logistics 48 (6), 484–495.
34. Teunter, R., 2004. Lot–sizing for inventory systems with product recovery. Computers & Industrial Engineering 46, 431–441.
35. Timothy, H.B., Dinesh, S.D., Kathy, E.F., Melvin, R.R., 1997. Economic lot size model for price-dependent demand under quantity and freight discounts. International Journal of Production Economics 48, 141–155.
36. Transchel, S., Minner, S., The impact of dynamic pricing on the economic order decision. European Journal of Operational Research 198, 773–789.
37. Webster, S., Mitra, S., 2007. Competitive strategy in remanufacturing and the impact of take-back laws. Journal of Operations Management 25, 1123–1140.
38. Zhao, X.F., Zhao, X.M., 2011. Pricing Decision and Performance Analysis of Closed-loop Supply Chain with Third-Party Reverse Logistics. Management and Service Science, 1–4.