在本篇論文中，我們研究著名的裝箱問題(the bin packing problem)¬¬以及此問題的延伸-三維空間的裝箱問題。此問題在實務上有非常多的應用，能有效的幫助許多產業，像是一般物流業、運輸業和製造業管理他們的存貨成本與運輸成本。因此裝箱問題的演算法設計與分析一直被廣為研究著。
為了使許多裝箱問題的演算法設計者有一個更好的參考工具可以衡量其演算法求解品質的好壞，我們針對此問題的下界值計算方法進行文獻探討與分析，也提出了最新三維空間裝箱問題的下界值並證明我們的結果改善了目前為止在文獻中已知的最佳結果，也就是Boschetti [3] 他們在2004年所提出的結果。另一方面，我們也提出了嚴格漸佳(Strictly better)的例子說明我們的下界值表現最好的部分，並且證明了我們下界值的最壞情況下成效比(Worst case performance ratio)，也就是在最糟糕的情況下，我們的下界值與最佳解的差距可證明在一個常數倍數內。
此外，我們也研究了裝箱問題的另一種延伸模型，也就是允許物件旋轉90度的模型，其下界值的計算方式，並同時證明我們提出的下界值，改善了文獻中，有關此模型最好的下界值。

In this thesis, we consider the three-dimensional orthogonal bin packing problem, which is a generalization of the well-known bin packing problem. The problem is useful to many practical applications, such as logistics industry, transportation industry and manufacturing industry. A better packing result can obviously reduce inventory costs and transportation costs.
In order to evaluate the quality of algorithms for the bin packing problem, we study the literatures and look into lower bounds of the problem. We present new low-er bounds for this problem and demonstrate that they improve the best previous results, the results proposed by Boschetti [3] in 2004. We also provide a strictly better example to emphasize the difference between the lower bounds. The asymptotic worst-case performance ratio, which can show that our lower bounds are bounded within a constant ratio in the worst case, is also proved.
In addition, we study an extension of this problem, the non-oriented model, which allows items to be rotated in 90 degrees. We study how to compute lower bounds for this model and demonstrate that our lower bounds improve the best previ-ous results for this model.

摘要 i
Abstract ii
致謝 iii
目錄 iv
圖目錄 vi
第一章 緒論 1
1.1研究背景與動機 1
1.2研究目的 1
1.3研究方法 2
第二章 文獻探討 3
2.1裝箱問題定義與其應用面概述 3
2.2裝箱問題近似演算法文獻探討 4
2.3裝箱問題下界值文獻探討 5
第三章 方法論 7
3.1基礎工具與概念介紹 7
3.1.1一維空間裝箱問題之下界值探討 7
3.1.2對偶可行解函數之方法探討 8
3.2三維空間裝箱問題之下界值改善 9
3.3三維空間裝箱問題之新下界值 13
3.3.1最壞情況下成效比分析 20
3.4 允許物件旋轉之模型 26
第四章 結論與未來展望 32
4.1結論 32
4.2未來展望 32
參考文獻 33
附錄A 36
A.1下界值程式實驗結果 36

1. N. Bansal, M. Sviridenko. New Approximability and Inapproximability results for 2-dimensional bin packing. In Proc. 15th Symposium on Discrete Algorithms (SODA 04), 196-203, 2004.
2. N. Bansal, A. Caprara, M. Sviridenko. A New Approximation Method for Set Covering Problems, with Applications to Multidimensional Bin Packing. SIAM Journal on Computing, 39(4):1256-1278, 2010.
3. M.A. Boschetti. New lower bounds for the three-dimensional finite bin packing problem. Discrete Applied Mathematics, 140:241-58, 2004.
4. JM. Bourjolly, V. Rebetez. An analysis of lower bounds procedures for the bin packing problem. Computers & Operations Research, 32:395-405, 2005.
5. J. Carlier, F. Clautiaux, A. Moukrim. New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation. Computers & Operations Research, 34:2223-2250, 2007.
6. J. Carlier, E. N’eron. Computing redundant resources for the resource constrained project scheduling problem. European Journal of Operational Research, 176(3):1452-1463, 2007.
7. A. Caprara. Packing 2-Dimensional Bins in Harmony. In Proc. 43rd IEEE Annu. Found. Comp. Sci. (FOCS 02), 490-499, 2002.
8. A. Caprara. Packing d-Dimensional Bins in d Stages. Mathematics of Operations Research, 33(1):203-215, 2008.
9. M. Chlebík, J. Chlebíková. Inapproximability results for orthogonal rectangle packing problems with rotations, In Proc. 6th Italian Conference on Algorithms and Complexity (CIAC 06), 199-210, 2006.
10. F. Clautiaux, C. Alves, J.V. de Carvalho. A survey of dual feasible and superadditive functions. Annals Operation Research, 179:317-342, 2010.
11. F. Clautiaux, A. Jouglet, J.E. Hayek. A new lower bound for the non-oriented two-dimensional bin-packing problem. Operations Research Letters, 35:365-373, 2007.
12. F. Clautiaux, A. Moukrim, J. Carlier. New data-dependent dual-feasible functions and lower bounds for a two dimensional bin-packing problem. International Journal of Production Research, 47(2):537-560, 2009.
13. E.G. Coffman Jr., M.R. Garey, D.S. Johnson. Approximation algorithms for bin-packing: a survey. In D.S. Hochbaum (ed.) Approximaiton algorithms for NP-hard problems, 46-93, PWS Publishing, Boston MA, 1997.
14. J. R. Correa, C. Kenyon. Approximation Schemes for Multidimensional Packing. In Proc. 15th Symposium on Discrete Algorithms (SODA 04), 179-188, 2004.
15. T.G. Crainic, G. Perboli, M. Pezzuto, R. Tadei. Computing the asymptotic worst-case of bin packing lower bounds. European Journal of Operational Re-search, 183:1295-1303, 2007.
16. M. Dell’Amico, S. Martello, D. Vigo. A lower bound for the non-oriented two-dimensional bin packing problem. Discrete Applied Mathematics, 118:13-24, 2002.
17. S.P. Fekete, J. Schepers. New classes of fast lower bounds for bin packing prob-lems. Mathematics Programming, 91:11-31, 2001.
18. S.P. Fekete, J. Schepers. A general framework for bounds for higher-dimensional orthogonal packing problems. Mathematical Methods of Operations Research, 60:311-29, 2004.
19. W. Fernandez de la Vega, G.S. Lueker. Bin-packing can be solved within 1+ε in linear time. Combinatorica, 1:349-355, 1981.
20. M.R. Garey, D.S. Johnson. Computers and intractability: a guide to the theory of NP- completeness. W.H. Freeman and Co., New York, 1979.
21. D.S. Johnson. Near-optimal bin packing algorithms. Dissertation, Massachusetts Insititute of Technology, Cambridge, Massachusetts, 1973.
22. N. Karmarkar, R.M. Karp. An efficient approximation scheme for the one-dimensional bin packing problem. In Proc. 23rd IEEE Annu. Found. Comp. Sci. (FOCS 82), 312-320, 1982.
23. B.H. Korte, J. Vygen. Combinatorial Optimization Theory and Algorithms (Chap-ter 18). Springer, 2008.
24. M. Labbe, G. Laporte, Mercure. Capacitated vehicle routing on trees. Operations Research, 39:616-22, 1991.
25. J. Y-T. Leung, T.W. Tam, C.S. Wong, G.H. Young, F. Y.L. Chin. Packing squares into a square. Journal of Parallel and Distributed Computing, 10(3): 271-275, 1990.
26. G.S. Lueker. Bin packing with items uniformly distributed over intervals [a,b]. In Proc. 24th IE- EE Annu. Found. Comp. Sci. (FOCS 83), 289-297, 1983.
27. A. Lodi, S. Martello, M. Monaci. Two-dimensional packing problems: A survey. European Journal of Operational Research, 141: 241-252, 2002.
28. S. Martello, D. Pisinger, D. Vigo. The three-dimensional bin packing problem. Operations Research, 48(2):256-267, 2000.
29. S. Martello, D. Vigo. Exact solution of the two-dimensional finite bin packing problem. Management Science, 44:388-99, 1998.
30. S. Martello, P. Toth. Knapsack problems: algorithms and computer implementa-tions. John Wiley & Sons, Chichester, U.K., 1990.
31. S. Martello, P. Toth. Lower bounds and reduction procedures for the bin packing problem. Discrete Applied Mathematics, 28:59-70, 1990.
32. J. Rietz, C. Alves, J.M. Valerio. Theoretical investigations on maximal dual feasible functions. Operations Research Letters, 38:174-178, 2010.
33. A. Scholl, R.Klein, C. Jurgens. BISON: A fast hybrid procedure for exactly solv-ing the one-dimensional bin packing problem. Computers & Operations Research, 24:627-645, 1997.
34. S.S. Seiden, R. van Stee. New bounds for multi-dimensional packing. Algorithmica, 36: 261-293, 2003.
35. V.V. Vazirani. Approximation algorithms. Springer-Verlag, Berlin, 2001.