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2019, Volume 15, Issue 3: 1213-1223. Doi: 10.3934/jimo.2018092
This issue Previous Article Maritime inventory routing problem with multiple time windows Next Article A hybrid inconsistent sustainable chemical industry evaluation method

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

    * Corresponding author: Yibing Lv 
Received:  May 2017
Revised:  October 2017
Early access:  June 2018
Published:  April 2019
The first author is supported by the National Natural Science Foundation of China grant 11771058, 11201039, 71471140, 91647204.
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    Abstract

  • 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.

    Mathematics Subject Classification: Primary: 90C26; Secondary: 90C30.

    Citation:
    shu

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  • 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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    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 $(x^{*}, y^{*})=(70.0,100.0, 70.0)$ $(x^{*}, y^{*})=(70.0,100.0, 70.0)$
    $F(x^{*}, y^{*})=(10.0, 5.0)$ $F(x^{*}, y^{*})=(10.0, 5.0)$
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