Solving Bilevel Linear Multiobjective Programming Problems

ABSTRACT

This study addresses bilevel linear multi-objective problem issues i.e the special case of bilevel linear programming problems where each decision maker has several objective functions conflicting with each other. We introduce an artificial multi-objective linear programming problem of which resolution can permit to generate the whole feasible set of the upper level decisions. Based on this result and depending if the leader can evaluate or not his preferences for his different objective functions, two approaches for obtaining Pareto- optimal solutions are presented.

This study addresses bilevel linear multi-objective problem issues i.e the special case of bilevel linear programming problems where each decision maker has several objective functions conflicting with each other. We introduce an artificial multi-objective linear programming problem of which resolution can permit to generate the whole feasible set of the upper level decisions. Based on this result and depending if the leader can evaluate or not his preferences for his different objective functions, two approaches for obtaining Pareto- optimal solutions are presented.

KEYWORDS

Multiobjective Programming, Bilevel Programming, Feasible Solution, Pareto-Optimal Solutions

Multiobjective Programming, Bilevel Programming, Feasible Solution, Pareto-Optimal Solutions

Cite this paper

nullC. Pieume, P. Marcotte, L. Fotso and P. Siarry, "Solving Bilevel Linear Multiobjective Programming Problems,"*American Journal of Operations Research*, Vol. 1 No. 4, 2011, pp. 214-219. doi: 10.4236/ajor.2011.14024.

nullC. Pieume, P. Marcotte, L. Fotso and P. Siarry, "Solving Bilevel Linear Multiobjective Programming Problems,"

References

[1] B. Colson, P. Marcotte and G. Savard, “An Overview of Bilevel Optimization,” Annals of Operational Research, Vol. 153, No. 1, 2007, pp. 235-256. doi:10.1007/s10479-007-0176-2

[2] J. Fulop, “On the Equi-valence between a Linear Bilevel Programming Problem and Linear Optimization over the Efficient Set,” Technical Report WP93-1, Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences, 1993.

[3] C. O. Pieume, L. P. Fotso and P. Siarry, “A Method for Solving Bilevel Linear Program-ming Problem,” Journal of Information and Optimization Science, Vol. 29, No. 2, 2008, pp. 335-358.

[4] Y. Yin, “Multiobjective Bilevel Optimization for Transportation Planning and Management Problems,” Journal of Advanced Trans-portation, Vol. 36, No. 1, 2000, pp. 93-105. doi:10.1002/atr.5670360106

[5] G. Eichfelder, “Multiobjective Bilevel Optimization,” Mathematical Programming, Vol. 123, No. 2, 2008, pp. 419-449. doi:10.1007/s10107-008-0259-0

[6] D. Kalyanmoy and S. Ankur, “Solving Bilevel Multi-Ob- jective Optimization Prob-lems Using Evolutionary Algorithms,” KanGAL Report Num-ber 2008005, 2008.

[7] I. Nishizaki and M. Sakawa, “Stack-elberg Solutions to Multiobjective Two-Level Linear Pro-gramming Problems,” Journal of Optimization Theory and Applications, Vol. 103, No. 1, 1999, pp. 161-182. doi:10.1023/A:1021729618112

[8] X. Shi and H. Xia, “In-teractive Bilevel Multi-Objective Decision Making,” Journal of the Operational Research Society, Vol. 48, No. 9, 1997, pp. 943-949.

[9] A. Messac and C. A. Mattson, “Generating Well Distributed Sets of Pareto Points for Engineering Using Physical Programming,” Optimization and Engineering, Vol. 3, No. 4, 2002, pp. 431-450. doi:10.1023/A:1021179727569

[10] C. A. Mattson, A. A. Mullur and A. Messac, “Smart Pareto Filter: Obtaining a Mi-nimal Representation of Multiobjective Design Space,” Engi-neering Optimization, Vol. 36, No. 6, 2004, pp. 721-740. doi:10.1080/0305215042000274942

[11] S. Sayin, “A Proce-dure to Find Discrete Representation of the Efficient Set with Specified Cover Errors,” Operations Research, Vol. 51, No. 3, 2003, pp. 427-436. doi:10.1287/opre.51.3.427.14951

[12] H. P. Benson, “An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set,” Journal of Global Optimization, Vol. 1, No. 1, 1991, pp. 83-104. doi:10.1007/BF00120667

[13] Y. Yamamoto, “Optimization over the Efficient Set: Overview,” Journal of Global Optimiza-tion, Vol. 22, No. 1-4, 2002, pp. 285-317. doi:10.1023/A:1013875600711

[14] C. O. Pieume, L. P. Fotso and P. Siarry, “Finding Efficient Set in Multiobjective Linear Programming,” Journal of Information and Optimization Science, Vol. 29, No. 2, 2008, pp. 203-216.

[15] M. H. Farahi and E. Ansari, “A New Approach to Solve Multi-Objective Linear Bilevel Programming Problems,” Journal of Mathematics and Computer Science, Vol. 1, No. 4, 2010, pp. 313-320.

[1] B. Colson, P. Marcotte and G. Savard, “An Overview of Bilevel Optimization,” Annals of Operational Research, Vol. 153, No. 1, 2007, pp. 235-256. doi:10.1007/s10479-007-0176-2

[2] J. Fulop, “On the Equi-valence between a Linear Bilevel Programming Problem and Linear Optimization over the Efficient Set,” Technical Report WP93-1, Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences, 1993.

[3] C. O. Pieume, L. P. Fotso and P. Siarry, “A Method for Solving Bilevel Linear Program-ming Problem,” Journal of Information and Optimization Science, Vol. 29, No. 2, 2008, pp. 335-358.

[4] Y. Yin, “Multiobjective Bilevel Optimization for Transportation Planning and Management Problems,” Journal of Advanced Trans-portation, Vol. 36, No. 1, 2000, pp. 93-105. doi:10.1002/atr.5670360106

[5] G. Eichfelder, “Multiobjective Bilevel Optimization,” Mathematical Programming, Vol. 123, No. 2, 2008, pp. 419-449. doi:10.1007/s10107-008-0259-0

[6] D. Kalyanmoy and S. Ankur, “Solving Bilevel Multi-Ob- jective Optimization Prob-lems Using Evolutionary Algorithms,” KanGAL Report Num-ber 2008005, 2008.

[7] I. Nishizaki and M. Sakawa, “Stack-elberg Solutions to Multiobjective Two-Level Linear Pro-gramming Problems,” Journal of Optimization Theory and Applications, Vol. 103, No. 1, 1999, pp. 161-182. doi:10.1023/A:1021729618112

[8] X. Shi and H. Xia, “In-teractive Bilevel Multi-Objective Decision Making,” Journal of the Operational Research Society, Vol. 48, No. 9, 1997, pp. 943-949.

[9] A. Messac and C. A. Mattson, “Generating Well Distributed Sets of Pareto Points for Engineering Using Physical Programming,” Optimization and Engineering, Vol. 3, No. 4, 2002, pp. 431-450. doi:10.1023/A:1021179727569

[10] C. A. Mattson, A. A. Mullur and A. Messac, “Smart Pareto Filter: Obtaining a Mi-nimal Representation of Multiobjective Design Space,” Engi-neering Optimization, Vol. 36, No. 6, 2004, pp. 721-740. doi:10.1080/0305215042000274942

[11] S. Sayin, “A Proce-dure to Find Discrete Representation of the Efficient Set with Specified Cover Errors,” Operations Research, Vol. 51, No. 3, 2003, pp. 427-436. doi:10.1287/opre.51.3.427.14951

[12] H. P. Benson, “An All-Linear Programming Relaxation Algorithm for Optimizing over the Efficient Set,” Journal of Global Optimization, Vol. 1, No. 1, 1991, pp. 83-104. doi:10.1007/BF00120667

[13] Y. Yamamoto, “Optimization over the Efficient Set: Overview,” Journal of Global Optimiza-tion, Vol. 22, No. 1-4, 2002, pp. 285-317. doi:10.1023/A:1013875600711

[14] C. O. Pieume, L. P. Fotso and P. Siarry, “Finding Efficient Set in Multiobjective Linear Programming,” Journal of Information and Optimization Science, Vol. 29, No. 2, 2008, pp. 203-216.

[15] M. H. Farahi and E. Ansari, “A New Approach to Solve Multi-Objective Linear Bilevel Programming Problems,” Journal of Mathematics and Computer Science, Vol. 1, No. 4, 2010, pp. 313-320.