本研究根據兩種不同的整數規劃演算法發展多維參數敏感度分析，此演算法分別為Bell和Shapiro(1975)提出的迭代對偶法以及分枝界限法，兩種演算法求解過程中不斷加入限制式至原問題中，因此我們可以利用求解過程中得到的資訊建立對偶問題，並進一步計算定理所需的閾值。根據本研究所提出的敏感度分析定理，在主問題與對偶問題強對偶成立的條件下，當目標係數的變化量不小於此閾值，則最佳解維持不變。最後，我們分別將提出的理論應用於多維參數目標係數變動問題的數值實驗中，並比較結果及計算效率。

This study develops multi-parametric cost coefficient sensitivity analysis theorems for integer programs from two different algorithms: the iterative dual method proposed by Bell and Shapiro (1975) and the branch and bound algorithm. We develop post-optimality analysis theorems, which indicate that the optimal solution remains optimal if the change of cost coefficients is not less than the thresholds. The theorem implies that with information as byproduct of these two algorithms, the thresholds can be computed. We carry out numerical experiments with multi-parametric cost perturbation problems using two proposed theorems separately and compare the results and computational tractability.

Table of Contents iii
1. Introduction 1
2. Literature Review 3
3. Methodology 5
3.1 Transformation 5
3.2 Setup for Sensitivity Theorem 1 7
3.3 Sensitivity Analysis via Group Equation 8
3.4 Constraints Decomposition 12
3.5 Setup for Sensitivity Theorem 2 13
3.6 Sensitivity Analysis via Constraints Decomposition Method 13
4. Result 18
4.1 The Environment of Experiments 18
4.2 Numerical Examples 18
4.3 A Real Example 31
5. Conclusions 39
References 40
Appendix 1 44
Appendix 2 45

Bell, D.E., Shapiro, J.F. 1975. A finitely convergent duality theory for zero-one integer programming. IIASA RM-75-33, International Institute for Applied Systems Analysis, Laxenburg, Austria.
Bell, D.E., Shapiro, J.F., 1977. A convergent duality theory for integer programming. Operations Research 25, 419-434.
Berkhemer OA, Fransen PS, Beumer D, van den Berg LA, Lingsma HF, Yoo AJ,…, Dippel DW. (2015). A randomized trial of intraarterial treatment for acute ischemic stroke. N Engl J Med. 2015;372:11–20.
Blair, C., 1997. Integer and mixed-integer programming. In Tomas Gal and Harvey J. Greenberg (eds.) Advances in Sensitivity Analysis and Parametic Programming, New York: Springer Science+Business Media, 9-1:9-25.
Boland, N.L., Eberhard, A.C., 2015. On the augmented Lagrangian dual for integer programming. Mathematical Programming Series A 150, 491-509.
Bowman, V.J., 1972. Sensitivity analysis in linear integer programming. AIIE Transactions 4, 284-289.
Bruce C.V. Campbell, Peter J. Mitchell, Timothy J. Kleinig, Helen M. Dewey, Leonid Churilov, Nawaf Yassi,…, Stephen M. Davis. (2015). Endovascular therapy for ischemic stroke with perfusion-imaging selection. N Engl J Med. 2015;372:1009–18.
Conforti, M., Cornu´ejols, G., Zambelli, G., 2014. Integer programming. Heidelberg: Springer.
Cook, W., Gerards, A.M.H., Schrijver, A., Tardos, I., 1986. Sensitiveity theorems in integer linear programming. Mathematical Programming 34, 251-264.
Dawande, M.W., Hooker, J.N., 2000. Inference-based sensitivity analysis for mixed integer/linear programming. Operations Research 48, 623-634.
Dua, V., Pistikopoulos, E.N., 2000. An algorithm for the solution of multiparametric mixed integer linear programming problems. Annals of Operations Research 99, 123-139.
Garfinkel, R.S., Nemhauser, G.L., 1972. Integer Programming. New York: Wiley Interscience.
Geoffrion, A.M., Nauss, R., 1977. Parametric and postoptimality analysis in integer linear programming. Management Science 23, 453-466.
Guzelsoy, M., Ralphs, T.K., 2010. Integer programming duality. Manuscript.
Holm, S., Klein, D., 1984. Three methods for postoptimal integer linear programming. Mathematical Programming Study 21, 97-109.
Jenkins, L., 1990. Parametric methods in integer linear programming. Annals of Operations Research 27, 77-96.
Jensen, R.E., 1968. Sensitivity analysis and integer linear programming. The Accounting Review 43, 425-446.
K¨ohler, J., 1996. Sensitivity analysis in integer linear programming: the right hand side case. Manuscript
Kocay, W.L., Kreher, D.L., 2017. Graphs, Algorithms, and Optimization 2nd edition. Boca Raton: CRC Press, Chapman and Hall.
Korte, B., Vygen, J., 2012. Combinatorial Optimization: Theory and Algorithms 5th Edition. Berlin: Springer-Verlag.
Li, D., Sun, X.L., 2006. Towards strong duality in integer programming. Journal of Global Optimization 35, 255-282.
Moazzez, B., 2014. Subadditive approaches to mixed integer programming. Manuscript.
Nauss, R.M., 1975. Parametric integer programming. Manucript.
Nemhauser, G.L., Wolsey, L.A., 1988. Integer and Combinatorial Optimization. New York: John Wiley & Sons. Papadimitriou, C.H., Steiglitz, K., 1988. Combinatorial Optimization. New York: Dover Publications Incorporation.
Papadimitriou, C.H., Steiglitz, K., 1988. Combinatorial Optimization. New York: Dover Publications Incorporation.
Radke, M.A., 1975. Sensitivity analysis in discrete optimization. Manuscript.
Richard, J.-P.P., Dey, S.S., 2010. The group-theoretic approach in mixed integer programming. In Ju¨nger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulley blank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958-2008, Berlin: Springer, 727-801.
Schrage, L., Wolsey, L., 1985. Sensitivity analysis for branch and bound integer programming. Operations Research 33, 1008-1023.
Shapiro, J.F., 1977. Sensitivity analysis in integer programming. Annals of Discrete Mathematics 1, 467-477.
Shapiro, J.F., 1979. Mathematical Programming: Structures and Algorithms. New York: John Wiley and Sons.
Skorin-Napov, J., Granot, F., 1987. Non-linear integer programming: sensitivity analysis for branch and bound. Operations Research Letters 6, 269-274.
Thomas, R.R., 2005. The structure of group relaxations. Chapter 3 in K. Aardal,
G.L. Nemhauser and R. Weismantel (eds.) Handbooks in Operations Research and
Management Science 12 - Discrete Optimization, Amsterdam: North Holland, 123-
170.
Vanderbeck, F., Wolsey, L.A., 2010. Reformulation and Decomposition of Integer Programs. Chapter 13, 431-502 in M. Ju¨nger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, L. Wolsey (eds.), 50 Years of Integer Programming 1958-2008 From the Early Years to the State-of-the-Art. Berlin, Germany: Springer-Verlag.
Wallace, S.W., 2000. Decision making under uncertainty: is sensitivity analysis of any use? Operations Research 48, 20-25.
Walukiewicz, S., 1991. Integer Programming. Dordrecht: Springer Science+Business Media.
Wang, H.-F., Horng, J.-S., 1996. Structural approach to parametric analysis of an IP on the case of the right-hand side. European Journal of Operational Research 92, 148-156.
Wendell, R.E., 2004. Tolerance sensitivity and optimality bounds in linear programming. Management Science 50, 797-803.
Wendell, R.E., Chen, W., 2010. Tolerance sensitivity analysis: thirty years later. Croatian Operational Research Review 1, 12-21.
Wolsey, L.A., 1981. Integer programming duality: price functions and sensitivity analysis. Mathematical Programming 20, 173-195.
Xu, Y., 2003. Asymptotic strong determination in integer programming: quasi dual method. Mathematical Methods of Operations Research 57, 207-216.