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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, 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.

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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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[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.

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.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[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.

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)
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