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doi: 10.3934/jimo.2018092

## Linear bilevel multiobjective optimization problem: Penalty approach

 1 School of Information and Mathematics, Yangtze University, Jingzhou 434023, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Yibing Lv

Received  May 2017 Revised  October 2017 Published  July 2018

Fund Project: The first author is supported by the National Natural Science Foundation of China grant 11771058, 11201039, 71471140, 91647204

In this paper, we are interested by the linear bilevel multiobjective programming problem, where both the upper level and the lower level have multiple objectives. We approach this problem via an exact penalty method. Then, we propose an exact penalty function algorithm. Numerical results showing viability of the algorithm proposed are presented.

Citation: Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018092
##### References:
 [1] M. Abo-Sinna, A bilevel nonlinear multiobjective decision making under fuzziness, Journal of Operational Research Society of India, 38 (2001), 484-495. doi: 10.1007/BF03398652. [2] G. Anandalingam and D. J. White, A solution for the linear static Stackelberg problem using penalty function, IEEE Transactions Automatic Control, 35 (1990), 1170-1173. doi: 10.1109/9.58565. [3] Z. Ankhili and A. Mansouri, An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research, 197 (2009), 36-41. doi: 10.1016/j.ejor.2008.06.026. [4] J. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1. [5] H. P. Benson, Optimization over the efficient set, Journal of Mathematical Analysis and Applications, 98 (1984), 562-580. doi: 10.1016/0022-247X(84)90269-5. [6] H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach, Journal of Optimization Theory and Applications, 131 (2006), 365-382. doi: 10.1007/s10957-006-9150-4. [7] H. I. Calvete and C. Gale, Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics, 234 (2010), 950-959. doi: 10.1016/j.cam.2008.12.010. [8] B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256. doi: 10.1007/s10479-007-0176-2. [9] K. Deb and A. Sinha, Solving bilevel multi-objective optimization problems using evolutionary algorithms, Lecture Notes in Computer Science: Evolutionary Multi-criterion Optimization, 5467 (2009), 110-124. doi: 10.1007/978-3-642-01020-0_13. [10] S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002. [11] S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. [12] S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical Programming, 131 (2012), 37-48. doi: 10.1007/s10107-010-0342-1. [13] G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449. doi: 10.1007/s10107-008-0259-0. [14] Y. B. Lv, An exact penalty function approach for solving the linear bilevel multiobjective programming problem, Filomat, 29 (2015), 773-779. doi: 10.2298/FIL1504773L. [15] Y. B. Lv and Z. P. Wan, Solving linear bilevel multiobjective programming problem via exact penalty function approach, Journal or Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0780-7. [16] I. Nishizaki and M. Sakawa, Stackelberg solutions to multiobjective two-level linear programming problem, Journal of Optimization Theory and Applications, 103 (1999), 161-182. doi: 10.1023/A:1021729618112. [17] M. S. Osman, M. A. Abo-Sinna, A. H. Amer and O. E. Emam, A multilevel nonlinear multiobjective decision making under fuzziness, Applied Mathematics and Computation, 153 (2004), 239-252. doi: 10.1016/S0096-3003(03)00628-3. [18] C. O. Pieume, P. Marcotte and L. P. Fotso, Solving bilevel linear multiobjective programming problems, American Journal of Operations Research, (2011), 214-219. doi: 10.4236/ajor.2011.14024. [19] M. Sakawa and I. Nishizaki, Cooperative and Noncooperative Multi-Level Programming, Springer, Berlin, 2009. doi: 10.1007/978-1-4419-0676-2. [20] X. Shi and H. Xia, Interactive bilevel multiobjective decision making, Journal of Operations Research Society, 48 (1997), 943-949. [21] X. Shi and H. Xia, Model and interative algorithm of bilevel multiobjective decision making with multiple interconnected decision makers, Journal of Multi-Criteria Decision Analysis, 10 (2001), 27-34. [22] H. W. Tang and X. Z. Qin, Pratical Methods of Optimization, Dalian University of Technology Press, Dalian, China, 2004. [23] C. Teng, L. Li and H. Li, A class of genetic algorithms on bilevel multiobjective decision making problem, Journal of Systems Science and Systems Engineering, 9 (2000), 290-296. [24] L. Vicente and P. Calamai, Bilevel and multilevel programming: A bibligraphy review, Journal of Global Optimization, 5 (1994), 291-306. doi: 10.1007/BF01096458. [25] J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184. doi: 10.1287/moor.1100.0480. [26] G. Zhang, J. Lu and T. Dillon, Decentralized multi-objective bilevel decision making with fuzzy demands, Knowledge-Based Systems, 20 (2007), 495-507. doi: 10.1016/j.knosys.2007.01.003. [27] Y. Zheng and Z. Wan, A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 37 (2011), 207-219. doi: 10.1007/s12190-010-0430-7.

show all references

##### References:
 [1] M. Abo-Sinna, A bilevel nonlinear multiobjective decision making under fuzziness, Journal of Operational Research Society of India, 38 (2001), 484-495. doi: 10.1007/BF03398652. [2] G. Anandalingam and D. J. White, A solution for the linear static Stackelberg problem using penalty function, IEEE Transactions Automatic Control, 35 (1990), 1170-1173. doi: 10.1109/9.58565. [3] Z. Ankhili and A. Mansouri, An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research, 197 (2009), 36-41. doi: 10.1016/j.ejor.2008.06.026. [4] J. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1. [5] H. P. Benson, Optimization over the efficient set, Journal of Mathematical Analysis and Applications, 98 (1984), 562-580. doi: 10.1016/0022-247X(84)90269-5. [6] H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach, Journal of Optimization Theory and Applications, 131 (2006), 365-382. doi: 10.1007/s10957-006-9150-4. [7] H. I. Calvete and C. Gale, Linear bilevel programs with multiple objectives at the upper level, Journal of Computational and Applied Mathematics, 234 (2010), 950-959. doi: 10.1016/j.cam.2008.12.010. [8] B. Colson, P. Marcotte and G. Savard, An overview of bilevel optimization, Annals of Operations Research, 153 (2007), 235-256. doi: 10.1007/s10479-007-0176-2. [9] K. Deb and A. Sinha, Solving bilevel multi-objective optimization problems using evolutionary algorithms, Lecture Notes in Computer Science: Evolutionary Multi-criterion Optimization, 5467 (2009), 110-124. doi: 10.1007/978-3-642-01020-0_13. [10] S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002. [11] S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359. doi: 10.1080/0233193031000149894. [12] S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical Programming, 131 (2012), 37-48. doi: 10.1007/s10107-010-0342-1. [13] G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449. doi: 10.1007/s10107-008-0259-0. [14] Y. B. Lv, An exact penalty function approach for solving the linear bilevel multiobjective programming problem, Filomat, 29 (2015), 773-779. doi: 10.2298/FIL1504773L. [15] Y. B. Lv and Z. P. Wan, Solving linear bilevel multiobjective programming problem via exact penalty function approach, Journal or Inequalities and Applications, 2015 (2015), 12pp. doi: 10.1186/s13660-015-0780-7. [16] I. Nishizaki and M. Sakawa, Stackelberg solutions to multiobjective two-level linear programming problem, Journal of Optimization Theory and Applications, 103 (1999), 161-182. doi: 10.1023/A:1021729618112. [17] M. S. Osman, M. A. Abo-Sinna, A. H. Amer and O. E. Emam, A multilevel nonlinear multiobjective decision making under fuzziness, Applied Mathematics and Computation, 153 (2004), 239-252. doi: 10.1016/S0096-3003(03)00628-3. [18] C. O. Pieume, P. Marcotte and L. P. Fotso, Solving bilevel linear multiobjective programming problems, American Journal of Operations Research, (2011), 214-219. doi: 10.4236/ajor.2011.14024. [19] M. Sakawa and I. Nishizaki, Cooperative and Noncooperative Multi-Level Programming, Springer, Berlin, 2009. doi: 10.1007/978-1-4419-0676-2. [20] X. Shi and H. Xia, Interactive bilevel multiobjective decision making, Journal of Operations Research Society, 48 (1997), 943-949. [21] X. Shi and H. Xia, Model and interative algorithm of bilevel multiobjective decision making with multiple interconnected decision makers, Journal of Multi-Criteria Decision Analysis, 10 (2001), 27-34. [22] H. W. Tang and X. Z. Qin, Pratical Methods of Optimization, Dalian University of Technology Press, Dalian, China, 2004. [23] C. Teng, L. Li and H. Li, A class of genetic algorithms on bilevel multiobjective decision making problem, Journal of Systems Science and Systems Engineering, 9 (2000), 290-296. [24] L. Vicente and P. Calamai, Bilevel and multilevel programming: A bibligraphy review, Journal of Global Optimization, 5 (1994), 291-306. doi: 10.1007/BF01096458. [25] J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184. doi: 10.1287/moor.1100.0480. [26] G. Zhang, J. Lu and T. Dillon, Decentralized multi-objective bilevel decision making with fuzzy demands, Knowledge-Based Systems, 20 (2007), 495-507. doi: 10.1016/j.knosys.2007.01.003. [27] Y. Zheng and Z. Wan, A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 37 (2011), 207-219. doi: 10.1007/s12190-010-0430-7.
The Pareto optimal solution obtained in this paper
 Exam. No. The Pareto optimal solution $(x^{*}, y^{*})$ obtained in this paper Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
 Exam. No. The Pareto optimal solution $(x^{*}, y^{*})$ obtained in this paper Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
Results in this paper and that in [15]
 Exam. No. Results in this paper Results by the algorithm in [15] Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ $(x^{*}, y^{*})=(0.6, 2.4, 0, 0)$ $F(x^{*}, y^{*})=(3.521, 7.958)$ $F(x^{*}, y^{*})=(5.4, 4.2)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
 Exam. No. Results in this paper Results by the algorithm in [15] Exam.3 $(x^{*}, y^{*})=(2.479, 0.521, 3.479, 5.0)$ $(x^{*}, y^{*})=(0.6, 2.4, 0, 0)$ $F(x^{*}, y^{*})=(3.521, 7.958)$ $F(x^{*}, y^{*})=(5.4, 4.2)$ Exam.4 $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $(x^{*}, y^{*})=(144.2, 26.8, 2.97, 67.7, 0)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ $F(x^{*}, y^{*})=(482.7, 1831.4)$ Exam.5 $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $(x^{*}, y^{*})=(11.938, 0, 0, 14.177, 2.786, 6.036)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ $F(x^{*}, y^{*})=(-364.008, -182.004)$ Exam.6 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
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