# American Institute of Mathematical Sciences

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:

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##### References:
Sets $f_{U}(x_i)$ of objective values of $x_i$, $i = 1,\ldots,5$
Sets ${\mbox{Min}}f_{U}(x_{i})$, ${\mbox{Min}}f_{U}(x_{i})-\mathbb{R}^{2}_\geqq$ and ${\mbox{Min}}f_{U}(x_{i})+\mathbb{R}^{2}_\geqq$, $i = 1,\ldots,5$
Sets ${\mbox{Max}}f_{U}(x_{i})$, ${\mbox{Max}}f_{U}(x_{i})-\mathbb{R}^{2}_\geqq$ and ${\mbox{Max}}f_{U}(x_{i})+\mathbb{R}^{2}_\geqq$, $i = 1,\ldots,5$
Sets $f_{U}(x^i)$, ${\mbox{Max}}f_{U}(x^i)$ and ${\mbox{Min}}f_{U}(x^i)$ of objective values of $x^i$, $i = 1,2$
Sets $f_{U}(x_i)$, ${\mbox{Max}}f_{U}(x_i)$ and ${\mbox{Min}}f_{U}(x_i)$ of objective values of $x_i$, $i = 1,2$
Sets $f_{U}(x_i)$, ${\mbox{Max}}f_{U}(x_i)$ and ${\mbox{Min}}f_{U}(x_i)$ of objective values of $x_i$, $i = 3,4$
Relationships between new concepts and the existent concepts of efficiency
Objective values of Table 1
Comparisons of solutions
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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