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An Alternative Approach for Solving Bi-Level Programming Problems
Department of Mathematics, Sir Padampat Singhania University, Bhatewar, India
Department of Mathematics, Mody University of Science and Techonology Lakshmangarh, India
Department of Mathematics, Integral University, Lucknow, India
Applied Mathematics & Humanities Department, Sardar Vallabhbhai National Institute of Technology, Surat, India
- 1 Department of Mathematics, Sir Padampat Singhania University, Bhatewar, India
- 2 Department of Mathematics, Mody University of Science and Techonology Lakshmangarh, India
- 3 Department of Mathematics, Integral University, Lucknow, India
- 4 Applied Mathematics & Humanities Department, Sardar Vallabhbhai National Institute of Technology, Surat, India
American Journal of Operations Research·Volume 07 (2017)·Pages 239–247·Published 4 May 2017·DOI10.4236/ajor.2017.73016
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Abstract
An algorithm is proposed in this paper for solving two-dimensional bi-level linear programming problems without making a graph. Based on the classification of constraints, algorithm removes all redundant constraints, which eliminate the possibility of cycling and the solution of the problem is reached in a finite number of steps. Example to illustrate the method is also included in the paper.
KeywordsLinear Programming ProblemBi-Level Programming ProblemGraphAlgorithm
- Bracken, J. and McGill, J. (1973) Mathematical Programs with Optimization Problems in the Constraints. Operation Research, 21, 37-44. https://doi.org/10.1287/opre.21.1.37
- Candler, W. and Norton, R. (1977) Multilevel Programming and Development Policy. Technical Report 258, Word Bank Staff, Washington DC.
- Wen, U. and Hsu, S. (1992) Efficient Solution for the Linear Bilevel Programming Problems. European Journal of Operation Research, 62, 354-362.
- Jan, R. and Chern, M. (1994) Nonlinear Integer Bilevel Programming. European Journal of Operation Research, 72, 574-587.
- Liu, Y. and hart, S. (1994) Characterizing an Optimal Solution to the Linear Bilevel Programming Problem. European Journal of Operation Research, 73, 164-166.
- Liu, Y. and Spencert, T. (1995) Solving a Bilevel Linear Program When the Inner Decision Maker Controls Few Variables. European Journal of Operation Research, 81, 644-651.
- Stackelberg, H. (1952) The Theory of Market Economy. Oxford University Press, Oxford.
- Bialas, W.F. and Karwan, M.H. (1982) On Tow-Level Linear Optimization. IEEE Transactions on Automatic Control, AC-27, 211-214.
- Bard, J.F. and Falk, J.E. (1982) An Explicit Solution to the Multi-Level Programming Problem. Computers and Operations Research, 9, 77-100.
- Bialas, W.F. and Karwan, M.H. (1984) Two-Level Linear Programming. Management Science, 30, 1004-1020. https://doi.org/10.1287/mnsc.30.8.1004
- Bard, J. (1985) Geometric and Algorithm Development for Hierarchical Planning Problem. European Journal of Operation Research, 19, 372-383.
- Mishra, V.N. (2007) Some Problems on Approximations of Functions in Banach Spaces. PhD Thesis, Indian Institute of Technology, Roorkee, 247-667.
- Hansen, P., Jaumard, B. and Sarvard, G. (1992) New Branch-and-Bound Rules for Linear Bilevel Programming. SIAM Journal of Scientific and Statistical Computing, 13, 1194-1217. https://doi.org/10.1137/0913069
- Judice, J. and Faustino, A. (1994) The Linear Quadratic Bi-Level Programming Problem. INFOR, 32, 87-98. https://doi.org/10.1080/03155986.1994.11732240
- Ben-Ayed, O. (1993) Bilevel Linear Programming. Computers and Operation Research, 20, 485-501.
- Haurie, A., Savard, G. and White, J.C. (1990) A Note on an Efficient Point Algorithm for a Linear Two-Stage Optimization Problem. Operation Research, 38, 553-555. https://doi.org/10.1287/opre.38.3.553