April  2019, 15(2): 705-721. doi: 10.3934/jimo.2018066

Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

* Corresponding author: Chun-Rong Chen

Received  July 2016 Revised  March 2018 Published  June 2018

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant number: 11301567) and the Fundamental Research Funds for the Central Universities (Grant number: 106112015CDJXY100002)

In this paper, we propose three concepts of robust efficiency for uncertain multiobjective optimization problems by replacing set order relations with the minmax less order relation, the minmax certainly less order relation and the minmax certainly nondominated order relation, respectively. We make interpretations for these concepts and analyze the relations between new concepts and the existent concepts of efficiency. Some examples are given to illustrate main concepts and results.

Citation: Hong-Zhi Wei, Chun-Rong Chen. Three concepts of robust efficiency for uncertain multiobjective optimization problems via set order relations. Journal of Industrial & Management Optimization, 2019, 15 (2) : 705-721. doi: 10.3934/jimo.2018066
References:
[1]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009. doi: 10.1515/9781400831050. Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program., 88 (2000), 411-424. doi: 10.1007/PL00011380. Google Scholar

[3]

J. R. Birge and F. V. Louveaux, Introduction to Stochastic Programming, Springer, New York, 1997. Google Scholar

[4]

M. EhrgottJ. Ide and A. Schöbel, Minmax robustness for multi-objective optimization problems, European J. Oper. Res., 239 (2014), 17-31. doi: 10.1016/j.ejor.2014.03.013. Google Scholar

[5]

M. Ehrgott, Multicriteria Optimization, Springer, New York, 2005. Google Scholar

[6]

G. Eichfelder and J. Jahn, Vector optimization problems and their solution concepts, in Recent Developments in Vector Optimization (eds. Q. H. Ansari and J. C. Yao), Springer, Berlin, (2012), 1–27. doi: 10.1007/978-3-642-21114-0_1. Google Scholar

[7]

J. Fliege and R. Werner, Robust multiobjective optimization & applications in portfolio optimization, European J. Oper. Res., 234 (2014), 422-433. doi: 10.1016/j.ejor.2013.10.028. Google Scholar

[8]

P. Gr. GeorgievD. T. Luc and P. M. Pardalos, Robust aspects of solutions in deterministic multiple objective linear programming, European J. Oper. Res., 229 (2013), 29-36. doi: 10.1016/j.ejor.2013.02.037. Google Scholar

[9]

M. A. GobernaV. JeyakumarG. Li and J. Vicente-Pérez, Robust solutions to multi-objective linear programs with uncertain data, European J. Oper. Res., 242 (2015), 730-743. doi: 10.1016/j.ejor.2014.10.027. Google Scholar

[10]

J. Ide and E. Köbis, Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations, Math. Methods Oper. Res., 80 (2014), 99-127. doi: 10.1007/s00186-014-0471-z. Google Scholar

[11]

J. Ide and A. Schöbel, Robustness for uncertain multi-objective optimization: A survey and analysis of different concepts, OR Spectrum, 38 (2016), 235-271. doi: 10.1007/s00291-015-0418-7. Google Scholar

[12]

J. Jahn, Vector Optimization-Theory, Applications, and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6. Google Scholar

[13]

J. Jahn, Vectorization in set optimization, J. Optim. Theory Appl., 167 (2015), 783-795. doi: 10.1007/s10957-013-0363-z. Google Scholar

[14]

J. Jahn and T. X. D. Ha, New order relations in set optimization, J. Optim. Theory Appl., 148 (2011), 209-236. doi: 10.1007/s10957-010-9752-8. Google Scholar

[15]

V. JeyakumarG. M. Lee and G. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435. doi: 10.1007/s10957-014-0564-0. Google Scholar

[16]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals, Optimization, 62 (2013), 649-671. doi: 10.1080/02331934.2013.769104. Google Scholar

[17]

E. Köbis, On robust optimization: Relations between scalar robust optimization and unconstrained multicriteria optimization, J. Optim. Theory Appl., 167 (2015), 969-984. doi: 10.1007/s10957-013-0421-6. Google Scholar

[18]

E. Köbis, On Robust Optimization: A Unified Approach to Robustness Using a Nonlinear Scalarizing Functional and Relations to Set Optimization, Ph. D. thesis, Martin-Luther-University in Halle-Wittenberg, 2014.Google Scholar

[19]

L. S. KongC. J. YuK. L. Teo and C. H. Yang, Robust real-time optimization for blending operation of alumina production, J. Ind. Manag. Optim., 13 (2017), 1149-1167. doi: 10.3934/jimo.2016066. Google Scholar

[20]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400. doi: 10.1016/S0362-546X(01)00274-7. Google Scholar

[21]

D. Kuroiwa and G. M. Lee, On robust multiobjective optimization, Vietnam J. Math., 40 (2012), 305-317. Google Scholar

[22]

A. Schöbel, Generalized light robustness and the trade-off between robustness and nominal quality, Math. Methods Oper. Res., 80 (2014), 161-191. doi: 10.1007/s00186-014-0474-9. Google Scholar

[23]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21 (1973), 1154-1157. Google Scholar

[24]

X. K. SunX. J. LongH. Y. Fu and X. B. Li, Some characterizations of robust optimal solutions for uncertain fractional optimization and applications, J. Ind. Manag. Optim., 13 (2017), 803-824. doi: 10.3934/jimo.2016047. Google Scholar

[25]

F. WangS. Y. Liu and Y. F. Chai, Robust counterparts and robust efficient solutions in vector optimization under uncertainty, Oper. Res. Lett., 43 (2015), 293-298. doi: 10.1016/j.orl.2015.03.005. Google Scholar

[26]

X. ZuoC. R. Chen and H. Z. Wei, Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings, J. Ind. Manag. Optim., 13 (2017), 475-486. doi: 10.3934/jimo.2016027. Google Scholar

show all references

References:
[1]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009. doi: 10.1515/9781400831050. Google Scholar

[2]

A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program., 88 (2000), 411-424. doi: 10.1007/PL00011380. Google Scholar

[3]

J. R. Birge and F. V. Louveaux, Introduction to Stochastic Programming, Springer, New York, 1997. Google Scholar

[4]

M. EhrgottJ. Ide and A. Schöbel, Minmax robustness for multi-objective optimization problems, European J. Oper. Res., 239 (2014), 17-31. doi: 10.1016/j.ejor.2014.03.013. Google Scholar

[5]

M. Ehrgott, Multicriteria Optimization, Springer, New York, 2005. Google Scholar

[6]

G. Eichfelder and J. Jahn, Vector optimization problems and their solution concepts, in Recent Developments in Vector Optimization (eds. Q. H. Ansari and J. C. Yao), Springer, Berlin, (2012), 1–27. doi: 10.1007/978-3-642-21114-0_1. Google Scholar

[7]

J. Fliege and R. Werner, Robust multiobjective optimization & applications in portfolio optimization, European J. Oper. Res., 234 (2014), 422-433. doi: 10.1016/j.ejor.2013.10.028. Google Scholar

[8]

P. Gr. GeorgievD. T. Luc and P. M. Pardalos, Robust aspects of solutions in deterministic multiple objective linear programming, European J. Oper. Res., 229 (2013), 29-36. doi: 10.1016/j.ejor.2013.02.037. Google Scholar

[9]

M. A. GobernaV. JeyakumarG. Li and J. Vicente-Pérez, Robust solutions to multi-objective linear programs with uncertain data, European J. Oper. Res., 242 (2015), 730-743. doi: 10.1016/j.ejor.2014.10.027. Google Scholar

[10]

J. Ide and E. Köbis, Concepts of efficiency for uncertain multi-objective optimization problems based on set order relations, Math. Methods Oper. Res., 80 (2014), 99-127. doi: 10.1007/s00186-014-0471-z. Google Scholar

[11]

J. Ide and A. Schöbel, Robustness for uncertain multi-objective optimization: A survey and analysis of different concepts, OR Spectrum, 38 (2016), 235-271. doi: 10.1007/s00291-015-0418-7. Google Scholar

[12]

J. Jahn, Vector Optimization-Theory, Applications, and Extensions, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6. Google Scholar

[13]

J. Jahn, Vectorization in set optimization, J. Optim. Theory Appl., 167 (2015), 783-795. doi: 10.1007/s10957-013-0363-z. Google Scholar

[14]

J. Jahn and T. X. D. Ha, New order relations in set optimization, J. Optim. Theory Appl., 148 (2011), 209-236. doi: 10.1007/s10957-010-9752-8. Google Scholar

[15]

V. JeyakumarG. M. Lee and G. Li, Characterizing robust solution sets of convex programs under data uncertainty, J. Optim. Theory Appl., 164 (2015), 407-435. doi: 10.1007/s10957-014-0564-0. Google Scholar

[16]

K. KlamrothE. KöbisA. Schöbel and C. Tammer, A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals, Optimization, 62 (2013), 649-671. doi: 10.1080/02331934.2013.769104. Google Scholar

[17]

E. Köbis, On robust optimization: Relations between scalar robust optimization and unconstrained multicriteria optimization, J. Optim. Theory Appl., 167 (2015), 969-984. doi: 10.1007/s10957-013-0421-6. Google Scholar

[18]

E. Köbis, On Robust Optimization: A Unified Approach to Robustness Using a Nonlinear Scalarizing Functional and Relations to Set Optimization, Ph. D. thesis, Martin-Luther-University in Halle-Wittenberg, 2014.Google Scholar

[19]

L. S. KongC. J. YuK. L. Teo and C. H. Yang, Robust real-time optimization for blending operation of alumina production, J. Ind. Manag. Optim., 13 (2017), 1149-1167. doi: 10.3934/jimo.2016066. Google Scholar

[20]

D. Kuroiwa, On set-valued optimization, Nonlinear Anal., 47 (2001), 1395-1400. doi: 10.1016/S0362-546X(01)00274-7. Google Scholar

[21]

D. Kuroiwa and G. M. Lee, On robust multiobjective optimization, Vietnam J. Math., 40 (2012), 305-317. Google Scholar

[22]

A. Schöbel, Generalized light robustness and the trade-off between robustness and nominal quality, Math. Methods Oper. Res., 80 (2014), 161-191. doi: 10.1007/s00186-014-0474-9. Google Scholar

[23]

A. L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21 (1973), 1154-1157. Google Scholar

[24]

X. K. SunX. J. LongH. Y. Fu and X. B. Li, Some characterizations of robust optimal solutions for uncertain fractional optimization and applications, J. Ind. Manag. Optim., 13 (2017), 803-824. doi: 10.3934/jimo.2016047. Google Scholar

[25]

F. WangS. Y. Liu and Y. F. Chai, Robust counterparts and robust efficient solutions in vector optimization under uncertainty, Oper. Res. Lett., 43 (2015), 293-298. doi: 10.1016/j.orl.2015.03.005. Google Scholar

[26]

X. ZuoC. R. Chen and H. Z. Wei, Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings, J. Ind. Manag. Optim., 13 (2017), 475-486. doi: 10.3934/jimo.2016027. Google Scholar

Figure 1.  Sets $f_{U}(x_i)$ of objective values of $x_i$, $i = 1,\ldots,5$
Figure 2.  Sets ${\mbox{Min}}f_{U}(x_{i})$, ${\mbox{Min}}f_{U}(x_{i})-\mathbb{R}^{2}_\geqq$ and ${\mbox{Min}}f_{U}(x_{i})+\mathbb{R}^{2}_\geqq$, $i = 1,\ldots,5$
Figure 3.  Sets ${\mbox{Max}}f_{U}(x_{i})$, ${\mbox{Max}}f_{U}(x_{i})-\mathbb{R}^{2}_\geqq$ and ${\mbox{Max}}f_{U}(x_{i})+\mathbb{R}^{2}_\geqq$, $i = 1,\ldots,5$
Figure 4.  Sets $f_{U}(x^i)$, ${\mbox{Max}}f_{U}(x^i)$ and ${\mbox{Min}}f_{U}(x^i)$ of objective values of $x^i$, $i = 1,2$
Figure 5.  Sets $f_{U}(x_i)$, ${\mbox{Max}}f_{U}(x_i)$ and ${\mbox{Min}}f_{U}(x_i)$ of objective values of $x_i$, $i = 1,2$
Figure 6.  Sets $f_{U}(x_i)$, ${\mbox{Max}}f_{U}(x_i)$ and ${\mbox{Min}}f_{U}(x_i)$ of objective values of $x_i$, $i = 3,4$
Figure 7.  Relationships between new concepts and the existent concepts of efficiency
Figure 8.  Objective values of Table 1
Figure 9.  Comparisons of solutions
Table 1.  Grades of the tourist spots in categories EF and TC
EF and TC $S_1$ $S_2$ $S_3$ $S_4$ $S_5$ $S_6$ $S_7$ $S_8$ $S_9$ $S_{10}$
Scenario 1 (12, 8) (15, 13) (15, 10) (13, 6) (15, 7) (14, 9) (14, 7) (9, 8) (16, 12) (14, 6)
Scenario 2 (9, 3) (15, 13) (10, 8) (6, 5) (7, 3) (8, 4) (8, 5) (7, 8) (13, 10) (7, 5)
Scenario 3 (4, 9) (15, 13) (10, 8) (4, 7) (3, 8) (5, 10) (9, 10) (10, 16) (13, 10) (5, 7)
Scenario 4 (10, 14) (15, 13) (13, 13) (6, 10) (7, 15) (8, 12) (5, 9) (17, 10) (15, 8) (10, 11)
EF and TC $S_1$ $S_2$ $S_3$ $S_4$ $S_5$ $S_6$ $S_7$ $S_8$ $S_9$ $S_{10}$
Scenario 1 (12, 8) (15, 13) (15, 10) (13, 6) (15, 7) (14, 9) (14, 7) (9, 8) (16, 12) (14, 6)
Scenario 2 (9, 3) (15, 13) (10, 8) (6, 5) (7, 3) (8, 4) (8, 5) (7, 8) (13, 10) (7, 5)
Scenario 3 (4, 9) (15, 13) (10, 8) (4, 7) (3, 8) (5, 10) (9, 10) (10, 16) (13, 10) (5, 7)
Scenario 4 (10, 14) (15, 13) (13, 13) (6, 10) (7, 15) (8, 12) (5, 9) (17, 10) (15, 8) (10, 11)
[1]

Yihong Xu, Zhenhua Peng. Higher-order sensitivity analysis in set-valued optimization under Henig efficiency. Journal of Industrial & Management Optimization, 2017, 13 (1) : 313-327. doi: 10.3934/jimo.2016019

[2]

Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

[3]

Xiang-Kai Sun, Xian-Jun Long, Hong-Yong Fu, Xiao-Bing Li. Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. Journal of Industrial & Management Optimization, 2017, 13 (2) : 803-824. doi: 10.3934/jimo.2016047

[4]

Guolin Yu. Global proper efficiency and vector optimization with cone-arcwise connected set-valued maps. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 35-44. doi: 10.3934/naco.2016.6.35

[5]

Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031

[6]

Nithirat Sisarat, Rabian Wangkeeree, Gue Myung Lee. Some characterizations of robust solution sets for uncertain convex optimization problems with locally Lipschitz inequality constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-25. doi: 10.3934/jimo.2018163

[7]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

[8]

Chunrong Chen, T. C. Edwin Cheng, Shengji Li, Xiaoqi Yang. Nonlinear augmented Lagrangian for nonconvex multiobjective optimization. Journal of Industrial & Management Optimization, 2011, 7 (1) : 157-174. doi: 10.3934/jimo.2011.7.157

[9]

Giancarlo Bigi. Componentwise versus global approaches to nonsmooth multiobjective optimization. Journal of Industrial & Management Optimization, 2005, 1 (1) : 21-32. doi: 10.3934/jimo.2005.1.21

[10]

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

[11]

Liping Zhang, Soon-Yi Wu. Robust solutions to Euclidean facility location problems with uncertain data. Journal of Industrial & Management Optimization, 2010, 6 (4) : 751-760. doi: 10.3934/jimo.2010.6.751

[12]

Ai-Ling Yan, Gao-Yang Wang, Naihua Xiu. Robust solutions of split feasibility problem with uncertain linear operator. Journal of Industrial & Management Optimization, 2007, 3 (4) : 749-761. doi: 10.3934/jimo.2007.3.749

[13]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697

[14]

Kequan Zhao, Xinmin Yang. Characterizations of the $E$-Benson proper efficiency in vector optimization problems. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 643-653. doi: 10.3934/naco.2013.3.643

[15]

Marius Durea, Elena-Andreea Florea, Radu Strugariu. Henig proper efficiency in vector optimization with variable ordering structure. Journal of Industrial & Management Optimization, 2019, 15 (2) : 791-815. doi: 10.3934/jimo.2018071

[16]

Zutong Wang, Jiansheng Guo, Mingfa Zheng, Youshe Yang. A new approach for uncertain multiobjective programming problem based on $\mathcal{P}_{E}$ principle. Journal of Industrial & Management Optimization, 2015, 11 (1) : 13-26. doi: 10.3934/jimo.2015.11.13

[17]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[18]

Vadim Azhmyakov. An approach to controlled mechanical systems based on the multiobjective optimization technique. Journal of Industrial & Management Optimization, 2008, 4 (4) : 697-712. doi: 10.3934/jimo.2008.4.697

[19]

Chunyang Zhang, Shugong Zhang, Qinghuai Liu. Homotopy method for a class of multiobjective optimization problems with equilibrium constraints. Journal of Industrial & Management Optimization, 2017, 13 (1) : 81-92. doi: 10.3934/jimo.2016005

[20]

Truong Q. Bao, Boris S. Mordukhovich. Refined necessary conditions in multiobjective optimization with applications to microeconomic modeling. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1069-1096. doi: 10.3934/dcds.2011.31.1069

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (91)
  • HTML views (920)
  • Cited by (0)

Other articles
by authors

[Back to Top]