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July  2019, 15(3): 1213-1223. doi: 10.3934/jimo.2018092

Linear bilevel multiobjective optimization problem: Penalty approach

Yibing Lv 1,, and  Zhongping Wan 2,

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 2019 Early access  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.

Keywords: Linear bilevel multiobjective programming, multiple objective programming, pareto optimal solution, penalty methods, numerical results.
Mathematics Subject Classification: Primary: 90C26; Secondary: 90C30.
Citation: Yibing Lv, Zhongping Wan. Linear bilevel multiobjective optimization problem: Penalty approach. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1213-1223. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002.  Google Scholar

[11]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.  Google Scholar

[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.  Google Scholar

[13]

G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449.  doi: 10.1007/s10107-008-0259-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Sakawa and I. Nishizaki, Cooperative and Noncooperative Multi-Level Programming, Springer, Berlin, 2009. doi: 10.1007/978-1-4419-0676-2.  Google Scholar

[20]

X. Shi and H. Xia, Interactive bilevel multiobjective decision making, Journal of Operations Research Society, 48 (1997), 943-949.   Google Scholar

[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.   Google Scholar

[22]

H. W. Tang and X. Z. Qin, Pratical Methods of Optimization, Dalian University of Technology Press, Dalian, China, 2004. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[25]

J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184.  doi: 10.1287/moor.1100.0480.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. Bard, Practical Bilevel Optimization: Algorithm and Applications, Kluwer, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Dempe, Foundations of Bilevel Programming, Kluwer, Dordrecht, 2002.  Google Scholar

[11]

S. Dempe, Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints, Optimization, 52 (2003), 333-359.  doi: 10.1080/0233193031000149894.  Google Scholar

[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.  Google Scholar

[13]

G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449.  doi: 10.1007/s10107-008-0259-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Sakawa and I. Nishizaki, Cooperative and Noncooperative Multi-Level Programming, Springer, Berlin, 2009. doi: 10.1007/978-1-4419-0676-2.  Google Scholar

[20]

X. Shi and H. Xia, Interactive bilevel multiobjective decision making, Journal of Operations Research Society, 48 (1997), 943-949.   Google Scholar

[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.   Google Scholar

[22]

H. W. Tang and X. Z. Qin, Pratical Methods of Optimization, Dalian University of Technology Press, Dalian, China, 2004. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[25]

J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Reseach, 36 (2011), 165-184.  doi: 10.1287/moor.1100.0480.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  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 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
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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 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
Table 2.  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 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
$F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
Table Options

Download as excel

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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